LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA 


Accession          o  b  1 9  8         Class 


COMPLIMENTS 

AMERICAN  BOOK  CO, 

A.  F.  GUNN,  Gen'l  Ag't, 
204  PINE  STRKET, 

SAN  FRANCISCO. 


GRAMMAR    SCHOOL 


ARITHMETIC 


BY 

A.    R.    HORNBROOK,   A.M. 

• 


NEW  YORK  •:•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


HORNBROOK'S    MATHEMATICS. 


HORNBROOK'S  PRIMARY  ARITHMETIC. 

Number  Studies  for  the  Second,  Third,  and  Fourth 
Years. 

HORNBROOK'S  GRAMMAR  SCHOOL  ARITHMETIC. 
A  Course  for  the  Last  Four  Years. 

HORNBROOK'S  CONCRETE  GEOMETRY. 
An  Introduction  to  Geometry. 


COPYRIGHT,  1900,  BY 
A.   R.   HORNBROOK. 

GUAM.   SCU.    ABITH. 

W.    P.    I 


DISTINCTIVE  FEATURES 

THIS  Arithmetic  is  designed  for  use  in  the  last  four  years 
of  the  grammar  schools.  The  method  of  presentation  is  the 
result  of  long  and  close  observation  in  the  schoolroom,  and 
conforms  to  the  order  and  manner  in  which  mathematical 
concepts  are  most  naturally  developed  in  children. 

Practical  work  has  been  so  combined  with  work  of  a  purely 
disciplinary  character  that  each  reenforces  and  enhances  the 
value  of  the  other.  In  business  arithmetic,  where  the  practi- 
cal demands  the  greater  emphasis,  the  most  simple  and  direct 
methods  of  computation  are  presented.  Applications  of  per- 
centage which  are  little  used  in  business,  but  which  have  a 
value  as  a  stimulus  to  thought,  are  introduced  at  the  point 
where  they  will  afford  the  best  discipline.  Some  subjects 
that  have  neither  a  practical  nor  a  high  disciplinary  value, 
though  found  in  many  text-books,  are  designedly  omitted  from 
this  book.  The  time  saved  by  the  omission  of  such  matter  is 
devoted  to  more  fruitful  drill  on  practical  exercises. 

A  carefully  planned  and  continuous  system  of  reviews  runs 
through  the  book.  These  reviews  take  the  form,  first,  of  an 
excursion  at  the  end  of  each  chapter  over  all  the  ground  thus 
far  traversed,  and,  second,  of  a  constant  correlation  of  acquired 
knowledge  with  concepts  about  to  be  developed. 

No  hard  and  fast  line  is  drawn  between  mental  and  written 
work.  Economy  of  time  and  effort  is  the  sole  basis  of  distinc- 
tion, and  this  is  a  self-regulating  principle. 

Rules  and  definitions  are  given  as  guides  in  the  preliminary 
stages  of  acquirement.  They  are  not  to  be  formally  memo- 

3 

86198 


4  DISTINCTIVE   FEATURES 

rized;  and,  when  clear  ideas  of  their  contents  have  been 
gained,  they  are  to  be  superseded  by  rules  and  definitions  of 
the  pupil's  own  framing. 

Constructive  work  with  simple  geometrical  forms  is  intro- 
duced at  intervals  whenever  the  numerical  relations  of  those 
forms  offer  valuable  material  illustrative  of  arithmetical  princi- 
ples. Exercises  to  test  and  develop  the  pupil's  power  of  visualiz- 
ing are  inserted  in  every  chapter.  The  pupil's  activity  is  further 
brought  into  play  by  a  series  of  exercises  in  which  he  is  called 
upon  to  supply  the  conditions  for  the  problems  as  well  as  their 
solution. 

Problems  involving  unknown  quantities,  which  are  solved 
arithmetically  only  by  most  complicated  processes,  are  deferred 
until  familiarity  with  some  of  the  principles  governing  the 
use  of  literal  quantities  may  suggest  simpler  methods  of 
procedure. 

The'  aim  throughout  has  been  to  secure  a  ready  skill  in  deal- 
ing with  numbers  and  to  develop  thought  power  adequate  to 
the  attack  of  any  arithmetical  problem  that  may  arise  in 
practical  life. 


CONTENTS 


CHAPTER  pAGE 

I.     INTEGERS  AND  DECIMALS       .......  7 

Fundamental  Operations  and  Proofs 16 

Addition  of  Decimals .27 

Subtraction  of  Decimals 30 

Multiplication  of  Decimals          ......  39 

Division  of  Decimals  ........  47 

Miscellaneous  Exercises 51 

II.     PROPERTIES  OP  NUMBERS       .......       62 

Multiples  and  Factors         .         ...         .  .62 

Composite  Numbers    ........       64 

Prime  Numbers  .........       65 

Prime  Factors 68 

Least  Common  Multiple 69 

Divisibility  of  Numbers       . 74 

Common  Divisors .77 

Powers  and  Roots 79 

Miscellaneous  Exercises 84 

III.  RATIO 95 

Miscellaneous  Exercises      .        .        .        .        .        .        .102 

IV.  FRACTIONS 108 

Addition  and  Subtraction  of  Fractions       .        .        .        .116 

Multiplication  of  Fractions 126 

Division  of  Fractions 134 

Miscellaneous  Exercises 142 

V.     DENOMINATE  NUMBERS 150 

Miscellaneous  Exercises      .......     185 

5 


6  CONTENTS 

CHAPTER  PAGE 

VI.     ALIQUOT  PARTS      .        .        . 199 

Miscellaneous  Exercises 210 

VII.     PERCENTAGE 216 

Merchandising 228 

Commission 230 

Trade  Discount 233 

Interest 236 

Promissory  Notes        .         .         .        .         .         .         .         .  244 

Partial  Payments 247 

Bank  Discount 250 

Insurance    ..........  254 

Taxes .259 

Miscellaneous  Exercises      .        .        .        ...        .        .  265 

VIII.     BONDS  AND  STOCKS 280 

Bonds 280 

Stocks 285 

Miscellaneous  Exercises 290 

IX.     LITERAL  QUANTITIES      ........  296 

Miscellaneous  Exercises • .  316 

X.     INVOLUTION  AND  EVOLUTION          ......  322 

Miscellaneous  Exercises 333 

XI.     PROPORTION 340 

Proportional  Parts 348 

Miscellaneous  Exercises 351 

XII.     MEASUREMENTS  AND  CONSTRUCTIONS     .....  .366 

Lines  and  Surfaces 366 

Solids.        .        .        .   ' 383 

Arcs  and  Angles 397 

Longitude  and  Time 408 

Miscellaneous  Exercises                                                         ,  412 


GRAMMAR   SCHOOL   ARITHMETIC 

CHAPTER   I 

INTEGERS   AND  DECIMALS 

1.  Write  an  integer  of  three  places. 

2.  Read:        235 

.  235  Read  "two  hundred,  thirty  five." 

2/o°r  Do   n°t    use   "and"   in   reading  an 


124,235 

3.  How  many  figures  are  used  to  express  the  last  number 
in  Ex.  2  ? 

4.  For  what  are  figures  used  ?    Explain. 

5.  Express  a  number  of  two  places  by  the  figures  5  and  3. 
Express  another  number  by  the  same  figures.     Which  is  the 
greater,  and  how  much  ? 

6.  What  is  the  largest  integer  that  can  be  expressed  by 
using  once  all  the  figures  3,  7,  and  5  ?     The  smallest  integer  ? 
Find  their  difference. 

NOTE  TO  TEACHER.      Strictly  speaking,  the  largest  integer  would  be 
53",  but  in  the  exercises  in  this  chapter  powers  of  numbers  are  excepted. 

7.  Find  the  difference  between  the  largest  integer  and  the 
smallest  integer  that  can  be  expressed  by  using  once  all  the 
figures  5,  1,  and  8. 

8.  There  are  six  different  integers  that  can  be  expressed 
by  using  once  all  the  figures  1,  2,  and  3.     Write  these  numbers 
in  the  order  of  their  size  and  find  their  sum. 

7 


8  INTEGERS   AND   DECIMALS 

9.    Can  numbers  be  expressed  without  figures  ? 

10.  Write  in  words  the  number  represented  by  105. 

11.  Express  in  good  English  the  number  represented  by 
228,427.     By  699,108. 

12.  Name  all  the  figures  that  are  used  to  express  number. 

13.  When  the  figure  0  stands  alone,  does  it  express  number? 
In  the  sentence  "  John  has  0  marbles,"  what  does  0  express  ? 

14.  0   is   called   naught,   zero,   or   cipher.     The   other   nine 
figures  used  to  express  numbers  in  Arabic  notation  are  called 
digits.     What  is  the  tens'  digit  of  the  number  75  ?     Of  235  ? 
What  is  the  thousands'  digit  of  the  number  8421  ?     Of  29834  ? 
Of  127446  ? 

15.  In  the  number  815,  which  is  greater,  the  hundreds'  digit 
or  the  tens'  digit  ?     How  much  ?     What  is  the  sum  of  all  the 
digits  of  that  number  ? 

16.  Write  a  number  the  sum  of  whose  digits  is  10. 

17.  Write  a  number  of  four  places  the  sum  of  whose  digits 
is  12. 

18.  Bead:          3 

30 
300  What  is  the  ratio  of  30  to  3  ?     Of 

o  00(]  300  to  30  ?     Of  each  number  in  the 

'  list  to  the  one  j  ust  before  it  ? 


300000 

SUGGESTION  TO  TEACHER.     If  pupils  are  not  familiar  with  the  terra 
"  ratio,"  substitute  the  question,  "  30  is  how  many  times  3  ?  " 

19.  Write  a  digit  and  place  0  at  the  right  of  it.     The  result 
equals  how  many  times  the  original  digit  ? 

20.  Placing  ciphers  at  the  right  of  a  digit  is  called  annexing 
ciphers  to  the  digit.     Annex  two  ciphers  to  5  and  state  how 
many  times  5  the  result  equals. 


INTEGERS   AND   DECIMALS  9 

21.  The  easiest  way  to  multiply  an  integer  by  10  is  to  annex 
one  cipher  to  it.     What  is  the  easiest  way  to  multiply  an  inte- 
ger by  100  ?     By  1000  ? 

22.  Give  at  sight  the  following  values  : 

a  35  multiplied  by  10     e   1000  times  16     e  10  times  30000 
b  100  times  71  d  10  times  3000    /  1000  times  50 

23.  Give  at  sight  the  quotient  of : 

a  40-hlO  c  4370 --10  e   15000  -r- 1000 

b  420  -T-  10  d  2500  H-  100  /  28000  -h  1000 

24.  Give  at  sight  the  following  values : 

a  TVof520  c  Ti^  of  2300  e  T^  of  4000 

b  T^  of  600        d  Ti¥  of  3100          /  T^TT  of  18000 

25.  CLASS  EXERCISE.     may  name  a  number  ending  in 

three  ciphers,  and  the  class  may  give  ^  of  it.     y-^  of  it. 

unr  <7  of  it- 

26.  Multiply  1000  by  1000.     A  thousand  thousands  equal  a 
Million.     How  many  figures  are  required  to  express  a  million  ? 

27.  Kead  8,636,448. 

Bead,  8  million  (not  millions),  636  thousand  (not  thousands), 
448. 

28.  Eead:  9,240,827.     31,676,201.     125,475,042. 

29.  Why  is  it  useful  to  separate  a  number  into  periods  of 
three  figures  each  before  reading  it  ? 

30.  Separate  into  periods  and  read:    8347621.      98470245. 
616823146.     47825001. 

31.  CLASS  EXERCISE.  -  may   write   9   figures   on   the 
board  in  a  horizontal  line,  and  another  pupil  may  tell  what 
number  they  represent. 

32.  Write  in  figures,  placing  a  comma  after  millions  and 
also  after  thousands :     5  million,  323  thousand,  471.     81  mil- 
lion, 175  thousand,  241.     815  million,  278  thousand,  924. 


10 


INTEGERS  AND   DECIMALS 


Millions 


Thousands 


Units 


00 

a 

-a 

C 

1 

CO 

0 

00 

'S 

1 
1 

73 

c 

o 

s 

• 
a 

o 

! 

T3 

1 

I 

00 

•TJ 

a 

o3 

CO 
£j 

w 

a> 

C 
3 

g 

H 

1 

a 
s 

1 

O 

H 

3                   §                    'S 

W              H               & 

4 

7 

6 

8 

2 

1 

023 

33.  CLASS  EXERCISE.     Copy  on  the  board  the  above  dia- 
gram, placing  different  figures  in  the  spaces  and  reading  the. 
numbers  thus  expressed. 

34.  Write  and  read  a  number  of  seven  places,  having  3  in 
the  millions'  place,  8  in  the  thousands'  place,  4  in  the  tens' 
place,  and  0  in  all  the  other  places. 

35.  Write  and  read  a  number  of  8  places,  having  2  in  the 
ten-millions'  place,  7  in  the  millions'  place,  4   in   the  units' 
place,  and  0  in  the  other  places. 

36.  When  numbers  are  expressed  in  figures  they  are  said  to 
be  written  in  Arabic  Notation.     Write  in  Arabic  notation  : 

a  323  million,  224  thousand,  24 

b  27  million,  960  thousand,    7 

c  169  million,  201  thousand,  25 

d  41  million,    41  thousand,  41 

e  75  million,    75  thousand,  76 

/  121  million,      3  thousand,    3 

37.  Write  a  number  of  7  places  whose  units'  figure  is  5. 
Find  $  of  it.     |  of  it.     -|-  of  it. 


INTEGERS  AND  DECIMALS  11 

38.  Write  the  largest  number  that  can  be  written  with  4 
places.     With  6  places.     With  9  places.      Give  the  sum  of 
the  digits  of  each  of  them. 

39.  How  many  can  you  count  in  a  minute  ? 

SUGGESTION  TO  TEACHER.  Find  by  trial  the  rate  of  speed  at  which 
different  pupils  count,  timing  them  by  the  watch. 

40.  At  your  rate  of  counting,  how  many  could  you  count  in 
an  hour  ?     In  a  day  of  10  hours  ? 

41.  Mary   Wallace,   a    little    girl   living   in  Philadelphia, 
counted  75   in   a  minute.     At  that   rate,   how  many   whole 
minutes  would  it  take  her  to  count  a  million?     How  many 
whole  hours  ?    How  many  days  if  she  counted  10  hours  a  day  ? 

42.  CLASS  EXERCISE.     may  report  the  number  which 

he  can  count  in  a  minute.     The  class  may  find  how  many 
minutes  would  be  required  for  him  to  count  a  million  at  that 
rate.     How  many  whole  hours.      How'  many  days  of  10  hours 
each. 

43.  Write  in  Arabic  notation : 

1st.  435  million,  347  thousand,  526. 

2d.  The  number  that  is  2  million  greater  than  the  1st. 

3d.  The  number  that  is  3  thousand  less  than  the  2d. 

4th.  The  number  that  is  300  thousand  more  than  the  3d. 

5th.  The  number  that  is  3  more  than  the  4th. 

6th.  The  number  that  is  20  thousand  less  than  the  5th. 

7th.  The  number  that  is  30  million  more  than  the  6th. 

8th.  The  number  that  is  200  million  more  than  the  7th. 

9th.  The  number  that  is  40  more  than  the  8th. 

44.  Find  difference  between  1st  and  9th  number  in  Ex.  43. 

SUGGESTION  FOR  CLASS  EXERCISE.  A  pupil  may  write  on  the  board  a 
number  containing  millions,  and  the  other  members  of  the  class  may 
direct  modifications  as  in  the  previous  examples.  When  the  pupil  at  the 
board  blunders,  another  pupil  may  take  up  his  work. 


12  INTEGERS  AND   DECIMALS 

45.  Beginning  at  2,  count  by  twos  to  10.     How  many  num- 
bers did  you  name  ? 

46.  What  is  the  sum  of  4  twos  ?    6  twos  ?    Numbers  which 
are  the  sum  of  a  number  of  twos  are  called  Even  Numbers. 

47.  What  is  the  first  even  number  after  20?     How  many 
twos  does  it  equal  ? 

48.  Write  all  the  even  numbers  that  can  be  expressed  by 
one  digit. 

49.  What  is  the  8th  even  number  ?     The  12th  even  number  ? 

50.  Divide  1,735,328  by  the  7th  even  number. 

51.  Can  you  write  an  even  number  which  does  not  end  with 
0,  or  2,  or  4,  or  6,  or  8  ? 

52.  Write  an  even  number  the  sum  of  whose  digits  is  9. 
Find  1  of  it.     Find  £  of  it.     Find  Jg-  of  it. 

•  s 

53.  Write  an  even  number  consisting  of  millions,  thousands, 
and  units.     Divide  that  number  by  32.     By  102.     By  104. 

54.  In  1895  the  expenses  of  the  United  States  government 
were  $  356,195,298.     The  revenues  of  the  government  for  that 
year  were  $  313,390,075.     How  much  did  the  amount  spent 
exceed  the  amount  received  ? 

55.  Mention  some  of  the  things  for  which  the  United  States 
government  spends  money,  and  make  an  example  in  addition. 

56.  The  cost  of  the  United  States  army  in  the  year  1895  was 
$51,804,759.     The  cost  of  the  navy  was  $28,797,796.     How 
much  did  they  both  cost  ? 

57.  Africa  contains  11,514,000  square  miles,  North  America 
6,446,000  square  miles,  South  America  6,837,000  square  miles, 
Asia  14,710,000  square  miles,  Australasia  3,228,000  square  miles, 
Europe  3,555,000  square  miles,  the  Polar  Kegions  4,888,800 
square  miles.     How  many  square  miles  of  land  does  the  whole 
world  contain  ? 


INTEGERS   AND   DECIMALS  13 

58.  The  total  exports  of  the  United  States  in  1895  amounted 
to  $  807,538,165 ;  the  imports  amounted  to  $  731,969,965.    How 
many  more  dollars'  worth  of  goods  were  sold  to  foreign  coun- 
tries than  were  bought  from  them  ? 

59.  The  earth  is  about  92,800,000  miles  from  the  sun;  the 
planet  Mars  about  140,000,000  miles  from  the  sun.    How  much 
nearer  to  the  sun  is  the  earth  than  Mars  ? 

60.  Multiply  a  million  by  a  thousand  by  annexing  ciphers. 

61.  A  thousand  millions  equal  a  Billion.    How  many  figures 
are  required  to  express  a  billion  ? 

62.  Point  off  and  read : 

a  414141414141  c  232648648648  e  58914367281 

b  673673673673  d   827345827345  /  42781632512 

63.  CLASS  EXERCISE.      may  write  twelve  figures  on 

the  board  in  a  horizontal  line,  and  others  may  tell  what  number 
they  represent. 

64.  Write  in  Arabic  notation  : 

1st.    427  billion,  338  million,  484  thousand,  521. 
2d.    The  number  that  is  4  billion  less  than  the  1st. 

3d.    The  number  that  is  2  billion,  7  million,  20  thousand  less 
than  the  2d. 

4th.  The  number  that  is  1  billion,  1  million,  and  1  thousand 
more  than  the  3d. 

5th.  The  number  that  is  13,013,013,013  more  than  the  4th. 

65.  CLASS  EXERCISE.     may  write  on  the  board  a  num- 
ber containing  billions,  and  the  class  may  direct  changes  of  it 
as  in  Ex.  64. 

66.  Write  and  read  an  even  number  consisting  of  billions, 
millions,  thousands,  and  units. 


14  INTEGERS  AND  DECIMALS 

67.  Write: 

a  98  billion,  348  million,  693  thousand,  207 

b  15  billion,  279  million,  427  thousand,  48 

c  216  billion,  849  million,  348  thousand,  7 

d  821  billion,  326  million,  475  thousand,  75 

e  2  billion,      2  million,      2  thousand,      2 

/  21  billion,    21  million,    21  thousand,  21 

g  78  billion,    78  million,    78  thousand,  78 

68.  CLASS  EXERCISE.     may  write  on  the  board  num- 
bers consisting  of  billions,  millions,  thousands,  and  units  which 
are  given  to  him  by  the  class. 

69.  To  count  a  billion  takes  how  many  times  as  long  as  to 
count  a  million  ? 

70.  From  the   time  of  the   establishment   of  our   govern- 
ment in  1789  till  1896  there  -had   been  spent   for  pensions 
$  1,950,403,063  and  for  interest  on  public  debts  $2,791,537,714. 
How  much  more  had  been  spent  for  interest  than  for  pensions  ? 
To  whom  are  pensions  given  ?     Why  ? 

71.  In   1881,  the   public   debt   of  the   United   States  was 
$  2,077,389,253  and  in  1882  it  was  $  1,926,688,678.     How  much 
was  the  debt  decreased  during  the  year  ? 

72.  In  1894,  Europe  produced  897,231,061  Ib.  of  wool,  North 
America  342,210,712  Ib.,  South  America  397,970,000  Ib.,  Cen- 
tral  America  2,000,000   Ib.,  Australia  663,600,000   Ib.,   Asia 
258,000,000  Ib.,  Africa  131,925,000  Ib.     How  many  pounds  of 
wool  were  produced  that  year  ? 

73.  Africa  has  about  127,000,000  inhabitants,  North  America 
89,250,000,  South  America  36,420,000,  Asia  850,000,000,  Aus- 
tralasia 4,730,000,  Europe  380,200,000,  Polar  Regions  300,000. 
What  is  the  entire  population  of  the  world  ? 

74.  Write  the  largest  number  that  can  be  written  with  12 
figures. 


INTEGERS  AND  DECIMALS  15 

75.  Write  a  number  of  15  places  and  find  from  the  follow- 
ing note  how  to  read  it. 

The  period  of  figures  next  higher  than  billions  is  called  trillions,  the 
next  quadrillions,  then  come  quintillions,  sextillions,  septillions,  octillions, 
nonillions,  decillions. 

76.  Write  a  number  larger  than  999  trillions  and  read  it. 
Why  do  we  seldom  use  such  large  numbers  ? 

77.  Write  the  largest  number  that  can  be  written  with  7 
places.     Find  1  of  it.     f  of  it.     ^T  of  it. 

78.  What  people  in  ancient  times  used   letters  to  express 
numbers  ?  . 

79.  Copy  the  Roman  numerals  and  write  under  each  the 
corresponding  Arabic  numeral. 

I        V        X        L        C        D        M 

80.  In  Eoman  notation  when  a  letter  is  repeated  its  value 
is  repeated.     Eead :  XX.     COG.     MMMM.     Write  in  Eoman 
notation:  3.     30.     300.     3000.     50.     500. 

81.  V,  L,  and  D  are  not  repeated.     Can  you  see  why  ? 

82.  When  a  letter  of  less  value  is  placed  after  a  letter  of 
greater  value  the  sum  of  their  values  is  represented.     Eead 
VIII.     XVI.    LXVI.    CLV.    MDCL.     MDCCCC.     Write  in 
Eoman  notation :   28.     36.     53.     75.     125.     381.     722.     1605. 
1620.     1905. 

83.  When  a  letter  of  less  value  is  placed  before  a  letter  of 
greater  value,   the  difference  of  their  values  is  represented. 
Eead :   IV.    IX.    XL.    XC.    MXCIX.     Write  in  Eoman  nota- 
tion: 14.    49.    99.    144.    579.    714.    1239.    1569.    1889.    1909. 

84.  A  line  over  a  letter  denotes  that  its  value  is  multiplied 
by  1000.     Eead:    MVI.     VDC.     Write   in  Eoman   notation: 
10051.     5525.     10630.     4324.     8956.     5427.     6385. 

85.  Write  the  following  numbers  in  Arabic  notation  and 
find  their  sum :    MDIII.    MDCCCIV.    MDCXX.     MDCCCCI. 
MMDLXV.     MDLXII.     MDCCCCIV.     MDCCCCXIX. 


16  INTEGERS   AND   DECIMALS 

86.  Write  the  following  numbers  in  Arabic   notation  and 
find  their  difference :  MCCXCIX  and  MDCCCXLV. 

87.  The  poet  Longfellow  was  born  in  MDCCCVII.     How 
many  years  old  was  he  at  the  breaking  out  of  the  Civil  War 
in  MDCCCLXI  ? 

88.  How  many  years  elapsed  between  July  4,  MDCCLXXVI, 
the  date  of  the  Declaration  of  Independence,  and  July  4th  of 
the  present  year  ? 

89.  Write  in  Arabic  notation,  MDC  and  find  y^-  of  it. 

90.  Write  in  Roman  notation  the  following  dates  : 
a  The  present  year. 

b  25  years  hence. 

c  100  years  before  you  were  born. 

d  The  year  in  which  our  present  president  was  elected. 

e  The  year  in  which  your  state  was  admitted  to  the  Union. 

/  The  year  of  Dewey's  victory  in  the  Philippine  Islands. 

SUGGESTION  FOR  CLASS  EXERCISE.     Let  the  pupils  suggest  important 
dates  to  be  written  in  Roman  notation  by  the  class. 

FUNDAMENTAL  OPERATIONS   AND   PROOFS 

91.  A  statement  that  two  quantities  are  equal  is  called  an 
Equation,  as  60  minutes  =  1  hour,  14  days  =  2  weeks,  8  -r-  4  =  2. 
Write  an  equation,  using  the  numbers,  7,  5,  and  another  number. 

92.  Numbers  that  are  added  are  called  Addends.     In  the 
equation  3  +  4  =  7,  which  numbers  are  addends  ? 

93.  Fill  out  the  following  equations  and  name  the  addends. 
5  +  7  =  ?     6+2i  =  ?     1+3  =  9 

94.  Give  two  addends  whose  sum  is  15.     27.     T8T. 

95.  Give  three  addends  whose  sum  is  14.     20.     -f. 

96.  Give  three  equal  addends  whose  sum  is  27.     30, 


FUNDAMENTAL  OPERATIONS   AND   PROOFS  17 

97.  When  John  has  caught  5  more  fish  he  will  have  caught 
7  fish.     How  many  has  he  caught  ? 

98.  If  Mr.  Reed  had  $  325  more,  he  could  buy  a  farm  cost- 
ing $  2168.     How  much  money  has  he  ? 

99.  When  the  sum  of  two  addends  is  29,  and  one  of  them 
is  4,  what  is  the  other  ?     Give  the  missing  addend  when  the 
sum  is  29,  and  the  known  addend  is  21. 

100.  CLASS  EXERCISE.     Think  of   two  addends   and   their 
sum.     Then  give  the  sum  and  one  of  the  addends  to  the  class. 
The  class  may  find  the  other  addend. 

101.  In  adding  25  and  18  James  carelessly  wrote  45  as  the 
answer.     If  either  of  his  addends  were  subtracted  from  the 
number  he  wrote,  would  the  other  addend  be  found  ?     Explain. 

102.  Add  13  and  26.     If  your  work  is  correct,  and  if  one 
addend  is  subtracted  from  your  answer,  what  will  be  left  ? 

103.  Complete  these  equations.     Illustrate  with  small  num- 
bers. 

Addend  -f-  Addend  =  Sum  —  Addend  = 

104.  Add  124  and  354  and  prove  your  work. 

To  prove  the  correctness  of  the  addition  of  two  numbers  subtract  one 
addend  from  their  sum.  If  the  work  is  right,  the  remaining  number  will 
equal  the  other  addend. 

105.  Find  sums  and  prove : 
abed  e  f 

628  949  639  457  1639  1854 

354  848  728  622  2136  237 

106.  Write   an    example   in   subtraction   and   show   which 
number  is  the  minuend.     The  subtrahend.     The  difference. 

107.  What  is  a  minuend ?     A  subtrahend?     A  difference? 

108.  From     728         Add  subtrahend  and  difference.     If  the 
take       516        work  is  correct,  the  result  will  equal  the 

minuend. 

HORN.    GRAM.    SCH.    AR. 2 


18  INTEGERS   AND   DECIMALS 

109.  From     824         Subtract  the  difference  from  the  minu- 
take      512        end.     If  the  work  is  correct,  the  result 

will  equal  the  subtrahend. 

110.  Complete  these  equations.     Illustrate. 

Minuend  —  Subtrahend  =  Mm.  —  Dif.  == 

Dif.  +  Sub.  = 

111.  Find  the  number  for  which  x  stands  in  the  following  : 

Minuend  Subtrahend  Difference 
a  240                              x  160 

b       x  16  30 

c     40  x  10 

d     60  50  x 

112.  CLASS  EXERCISE.     -  may  give  a  minuend  and  a 
difference.     The  class  may  find  the  subtrahend. 

SUGGESTION  TO  TEACHER.     Every  pupil  should  be  required  to  bring  to 
the  class  his  contribution  to  the  class  exercise  carefully  prepared. 

113.  CLASS  EXERCISE.     -  may  give  a  subtrahend  and  a 
difference,  and  the  class  may  find  the  minuend. 

114.  How  can  you  prove  an  example  in  subtraction  ? 

115.  Subtract  and  prove  : 

a  b  c  d  e  f  g 

849          623          814          338          599          451          2148 
321          517          276          124          378          239          1939 

116.  Multiply  123  by  3.     By  30.     By  300.     Find  the  sum 
of  the  products.     Compare  the  sum  of  the  products  with  the 
product  of  123  multiplied  by  333. 

117.  Multiply        275 
by  137 

In  this  example  of  what  two  num- 
bers  is  1925  the  product  ?    8250  ? 


275QO        27500  ?    How  is  37675  obtained  ? 


37675 


FUNDAMENTAL  OPERATIONS  AND  PROOFS  19 

SUGGESTION  TO  TEACHER.  Show  that  in  multiplying  by  a  number  of 
two  or  more  places  we  are  finding  the  sum  of  the  several  products  of  the 
multiplicand  and  the  number  expressed  by  each  figure  of  the  multiplier 
in  its  present  position  ;  and  that  in  practice  the  naughts  are  omitted  for 
the  sake  of  convenience. 

118.  Multiply  a  number  of  3  places  by  a  number  of  4  places, 
writing  out  the  full  partial  products.     Why  are  the  full  prod- 
ucts not  usually  written  out  ? 

119.  What  is  a  multiplicand  ?    A  multiplier  ?    A  product? 
Illustrate. 

120.  Use  25  as  a  multiplicand  and  17  as  a  multiplier. 

121.  If  the  product  of  25  and  17  is  divided  by  17,  what 
result  will  be  obtained  ?     If  the  product  of  25  and  17  is  divided 
by  25,  what  result  will  be' obtained  ? 

122.  When  the  product  of  two  numbers  is  divided  by  one 
of  the  numbers,  what  result  is  obtained  ?     Illustrate. 

123.  The  product  of  two  numbers  is  35,  the  multiplicand 
is  7.     Find  the  multiplier. 

124.  Take  Ex.  123,  substituting  in  turn  for  35  the  numbers 
84,  42,  77,  91,  112. 

125.  Find  the  values  of  x. 

Product  Multiplicand  Multiplier 
a  75                              x  15 

b     x  14  3 

c   70  x  10 

d  60  6  x 

126.  Complete  the  equations.    Illustrate. 
Multiplicand  x  Multiplier  =          Product  -r-  Multiplier  = 

Product  -r-  Multiplicand  = 

127.  How  can  you  prove  an  example  in  multiplication  ? 

128.  Multiply  and  prove : 

a  18  x  20  b  13  x  14  c  15  x  16  d  14  x  25 


20  INTEGERS  AND   DECIMALS 

129.  Multiply  and  prove : 

a  b  c  d  e  f 

836  457  791  625  927  654 

125  243  348  244  238  289 

130.  What   is    a   dividend?      A    divisor?      A    quotient? 
Illustrate. 

131.  Find  quotients  of  800 -f- 2,  60^2,  and  8-5-2.     Add 
the   quotients   and   compare  the   sum   with   the   quotient   of 
868  -r-  2. 

SUGGESTION  TO  TEACHER.     Show  that  in  dividing  868  by  2,  we  divide 
800  by  2,  then  60  by  2,  and  then  8  by  2. 

132.  Divide  by  3  first  900,  then  60,  and  then  6,  and  find  the 
sum  of  the  quotients.     Show  a  shorter  way  of  dividing  the 
sum  of  those  numbers  by  3. 

133.  Divide  400  and  80  and  4  each  by  4.     Show  the  usual 
way  of  dividing  the  sum  of  these  numbers  by  4. 

134.  764  -f-  2  =  ? 

In  dividing  764  by  2  we  divide  600  by  2,  then  160  by  2,  and  then  4  by  2. 

135.  976  -i-  4  =  ?     (800  -v-  4).+  (160  -  4)  +  (16  -h  4)  =  ? 

136.  Divide  765  by  5,  and  show  how  many  hundreds,  how 
many  tens,  and  how  many  units  are  used  in  the  separate 
divisions. 

137.  Divide  5468  by  4,  and  show  how  many  thousands,  how 
many  hundreds,  how  many  tens,  and  how  many  units  are  used 
in  the  divisions. 

138.  Divide  3765  by  5,  and  show  what  parts  of  the  number 
are  used  in  each  division. 

139.  What  number  divided  by  8  will  give  2  for  a  quotient  ? 

140.  The  quotient  of  a  certain  number  divided  by  8  is  7. 
What  is  the  dividend  ? 


FUNDAMENTAL  OPERATIONS   AND  PROOFS  21 

141.  Complete  the  following  equations.     Illustrate. 
Dividend  -+-  Divisor  =  Dividend  -*-  Quotient  = 

Divisor  x  Quotient  = 

142.  Find  the  values  of  x. 

Dividend         Divisor         Quotient         Dividend         Divisor         Quotient 
a  72  4  x  c     x  21  5 

b   99  x  11  d  32  x  4 

143.  Of  what  number  is  11  both   divisor  and   quotient  ? 
What  number  has  for  divisor  and  for  quotient  7  ?     12  ?     13  ? 
15?     17?    21? 

144.  Use  21  as  a  divisor  and  378  as  a  dividend.     Multi- 
ply divisor   by   quotient,   and   compare   the   result  with  the 
dividend. 

SUGGESTION  £0  TEACHER.  Using  small  numbers,  show  that  as  multi- 
plication and  division  are  reverse  processes,  they  may  be  used  to  prove 
each  other. 

145.  Make  a  rule  for  proving  an  example  in  division  when 
there  is  no  remainder. 

146.  Divide  each  of  the  following  numbers  by  23,  and  prove 
your  work : 

322  391  575  759  874  943  1288 

147.  Use  41  as  a  divisor  and  618  as  a  dividend.     Multiply 
divisor  by  quotient,  add  the  remainder,  and  compare  the  result 
with  the  dividend. 

148.  Make  a  rule  for  proving  an  example  in  division  when 
there  is  a  remainder. 

149.  Divide  each  of  the  following  numbers  by  25,  and  prove 
your  work : 

625         879         579         824         386         758         1028         981 


22  INTEGERS  AND  DECIMALS 

150.   Divide  26103  by  12  as  follows  : 

12)26103(2175 
24000 

2103 

1200  How   many  times   is   12    contained   in 

24,000  ?      How   much,   remains   to  be   di- 


vided    by   12    after   24,000   is   subtracted 

from  26,103? 
63 

60 


SUGGESTION  TO  TEACHER.  Show  the  process  of  dividing  each  re- 
mainder after  the  successive  subtractions,  and  call  attention  to  the  fact 
that  it  is  more  convenient  to  omit  ciphers  and  that  it  gives  the  same  result. 

151.  Divide  some  numbers  by  others  by  long  division,  writ- 
ing out  all  the  work.     Why  is  it  better  usually  to  omit  some 
of  the  work  ? 

152.  A  schoolboy  brought  this  example  to  his  teacher  and 

121)87493(723          told  her  that  he  had  discovered  that 
£.„  if  the  numbers  here  printed  in  heavy 

tvpe  were  added  in  the  order  in  which 

27Q 
'  *  they  stand,  the  result  would  equal  the 

*^*  dividend.     He  proved  his  problems  in 

373  long   division  in   that  way.      Take  an 

example  in  long  division  and  prove  it 
10  in  the  same  way. 

153.  Can  you  see  why  adding  all  the  subtrahends  and  the 
remainder   in   long  division  will  give  a  result  equal  to  the 
dividend  ? 

SUGGESTION  TO  TEACHER.  Show  that  in  this  example  700  times  121 
=  84,700,  20  times  121  =  2420,  3  times  121  =  363,  and  that  the  sum  of 
these  numbers,  plus  the  remainder  10  must  equal  the  dividend. 

154.  Divide  a  number  by  another  number  that  it  contains 
exactly  8  times.     Double  your  dividend  and  see  how  the  quo- 
tient is  changed. 


FUNDAMENTAL    OPERATIONS    AND    PROOFS  23 

155.  Work  with  small  numbers  and  show  the  truth  of  the 
following  principles : 

PRIN.  1.  Increasing  the  dividend  increases  the  quotient. 

PRIN.  2.  Decreasing  the  dividend  decreases  the  quotient. 

PRIN.  3.  Increasing  the  divisor  decreases  the  quotient. 

PRIN.  4.  Decreasing  the  divisor  increases  the  quotient. 

156.  Some  of  a  milkman's  customers  buy  a  pint  of  milk 
at  a  time,  and  some  buy  a  quart.      How  many  of  the  customers 
who  buy  a  pint  will  together  dispose  of  a  gallon  ?     How  many 
who  buy  a  quart  ? 

157.  How  many  pints  of  water  can  be  drawn  from  a  20- 
gallon  tank  ?     How  many  quarts  ?     How  many  gallons  ? 

158.  A  40-gallon  tank  contains  how  many  times  as  many 
pints  as  a  20-gallon  tank  ? 

159.  Illustrate  the  following  principles  with  small  numbers : 

PRIN.  5.  Multiplying  both  dividend  and  divisor  by  the  same 
number  does  not  change  the  quotient. 

PRIN.  6.  Dividing  both  dividend  and  divisor  by  the  same 
number  does  not  change  the  quotient. 

160.  Find  the  product  of  8  and  10  and  divide  that  product 
by  4. 

If   the   following  expression   of   the   problem   were   used, 

8  x  1Q,  and  if  before  multiplying,  both  8  and  4  were  divided 

2 
by  4  as  follows,  $  x  10  =  20,  would  the  result  be  the  same  ? 

F 

SUGGESTION  TO  TEACHER.  Let  pupils  prove  by  trial  with  many  small 
numbers  that  canceling  common  factors  in  dividend  and  divisor  does  not 
change  the  quotient. 


24  INTEGERS  AND  DECIMALS 

161.    Find  the  values  of  the  following,  canceling  when  you 
can: 

a  b  c 

4  x  8  x  6  _  6  x  9  x  18  21  x  4  x  6 


2x4x2  3x3x6~  7x2x3 

d  e  / 

40  x  8  x  10  3  x4  x  5  16  x 


=  9 
~ 


20x4x5  7x8x10  8x8 


21  x28x4  2  x  8  x  12  30  x  7  x  5 


_9  _  9 


7x7x6  5x4x6  60  x  21  x  15 

^  k  I 

80  x  4  x  25  =  ?  24  x  7  x  10  =  9  35  x  6  x  9     =  9 

16  x  20  x  5  12  x  14  x  20  18  x  7  x  5  x  3  " 

Can  you  show  how  the  process  of  cancellation  depends  upon 
Prin.  6  ? 

162.  Cancel  and  find  values : 

a  b  c 

5  x  8  x  10  x  27  7  x  9  x  56  x  65  21  x  84  x  6  x  8 

54x4x20  26  x  49  x~32  x  5  12x49x18 

d  e  f 

25  x  6  x  7  x  30  16  x  9  x  28  42  x  15  x  10  x  6 

50x9x5  8x36x2  25x14x5 

163.  CLASS  EXERCISE.     With  24  x  30  x  35  as  the  dividend, 
-  may  make  a  cancellation  exercise,  and  the  class  may 

solve  it. 

164.  CLASS   EXERCISE.     With   28  x  12  x  36    as   a  divisor, 
may  give  a  cancellation  exercise  for  the  class  to  solve. 

165.  Copy  upon  the  board  and  read  1,111,111. 

166.  Point  out  the  figure  that  expresses  a  hundred  thousand. 
How  many  hundred  thousand  make  a  million  ? 


FUNDAMENTAL    OPERATIONS    AND    PROOFS  25 

167.  What  part  of  a  million  is  a  hundred  thousand  ? 

168.  How  much  is  y1^  of  one  hundred  thousand?     Show 
the  figure  that  expresses  it. 

169.  How  much  is  y1^-  of  ten  thousand  ?     Show  the  figure 
that  stands  for  it. 

170.  Show  the  figure  that  stands  for  y^-  of  a  thousand,     y1^ 
of  a  hundred,     y1^  of  ten. 

171.  Copy  111.1.     The  point  after  the  units  is  called  a 
Decimal  Point. 

172.  The  mtmber  that  is  yL  of  1  is  written  .1.     Eead  111.1. 
This  should  be  read  111  and  1  tenth. 

173.  1  at  the  right  of  the  tenths'  place  means  y1^  of  yL  or 
yi¥.     How  many  tenths  and  hundredths  in  .11  ? 

174.  1  at  the  right  of  the  hundredths'  place  means  y^-  of 
j-^.  or  10100.     How  many  tenths,  hundredths,  and  thousandths 
in  .111  ? 

175.  What  does  1  at  the  right  of  the  thousandths'  place 
mean?    What  does  1  in  the  next  place  to  the  right  mean? 
In  the  next  ? 

176.  Numbers  written  at  the  right  of  the  decimal  point  are 
called  Decimals  or  Decimal  Fractions.     They  decrease  in  value 
at  a  tenfold  rate  from  left  to  right  just  as  integers  decrease. 
A  decimal  of  one  place  expresses  tenths.    What  does  a  decimal 
of  two  places  express  ?     Of  three  places  ? 

177.'  Eead: 

.1  .2  .9  .07  .007  .09  .009 

178.  Eead  .19. 

This  is  read  19  hundredths  (^%).  A  decimal  is  read  like  an  integer, 
and  then  the  name  of  the  last  decimal  place  is  added  to  show  what  kind 
of  fractional  parts  it  represents. 

179.  Eead: 

12.1      4.3     8.01      7.02      9.021      .003      .009      .03     .006 


26  INTEGERS   AND   DECIMALS 

180.  Write  and  read  a  decimal  of  two  places.     Of  three 
places.     Of  four  places. 

181.  Write  in  words  : 

.123    4.5      .4    41.41       .103    19.2 
.15     3.75    8.6    41.041    21.109    11.025 

182.  Eead: 

576.137         432.25         57.41         32.06        75.37        45.81 

183.  Write  in  one  number  an  integer  of  4  places  and  a 
decimal  of  3  places.     Read  it. 

184.  Write  and  read  a  number  consisting  of  an  integer  of 
6  places  and  a  decimal  of  2  places. 

185.  Write  and  read  a  number  in  which  there  is  an  integral 
part  of  3  places  and  a  decimal  part  of  2  places. 

186.  Write  and  read  a  number  in  which  there  is  an  integral 
part  of  5  places  and  a  decimal  part  of  3  places. 

187.  Which  is  greater  and  how  much,  the  integer  1  or  the 
fraction  ^  ? 

Fractions  whose  denominators  are  10,  100,  1000,  or  1  with  any  number 
of  ciphers  annexed,  may  be  written  as  decimals.  This  way  of  expressing 
such  fractions  is  convenient  because  when  thus  expressed  they  may  be 
added,  subtracted,  multiplied,  and  divided  in  the  same  way  as  integers. 

188.  Write  as  common  fractions  : 

.12        .029        .125        .17        .27        .013       .049       .019 

Notice  that  the  denominator  of  a  decimal  is  not  written,  but  is  indicated 
by  the  number  of  places  it  occupies.  It  is  always  1  with  as  many  ciphers 
annexed  as  there  are  places  in  the  decimal.  Thus  in  .32  the  denominator 
is  100. 

189.  Write  as  decimals: 


190.    Write  as  decimals  : 


A 


TTnT  A  TTOF  ifo  TTfr  T5~G 


ADDITION   OF   DECIMALS  27 

191.  Express  T9^,  using  only  one  figure.     Express   y^  by 
two  figures,     y-^  by  three  figures. 

192.  Write  as  common  fractions : 

.6        .06        .027        .125        .0004        .00025        .000005 

193.  If  you  cut  a  string  into  ten  equal  parts,  in  how  many 
places  must  you  cut  it  ?     What  is  each  part  called  ?     Express 
it  as  a  decimal. 

194.  How  ma^iy  times  must  a  string  be  cut  to  divide  it  into 
100  equal  parts  ?     Into  1000  equal  parts  ? 

195.  Show  on  a  ruler  .1  of  10  in.     .1  of  5  in. 

196.  If  20  pupils  were  in  a  class  and  .1  of  them  were  dis- 
missed, how  many  would  remain  ? 

197.  How  much  is  .1  of  100  ?     .01  of  100  ?     Why  can  you 
not  easily  show  on  a  ruler  .001  of  an  inch  ? 

ADDITION    OF    DECIMALS 

198.  Read  and  add: 

a  be 

462.001  321.12  725.375 

25.01  56.37  409.003 

63.475  81.07  361.1 

181.312  73.22  448.0035 

692.436  195.87  772.6 

199.  In  adding  decimals  why  is  it  best  to  arrange  them  so 
that  the  decimal  points  are  in  a  vertical  line  ? 

200.  Write  and  add  : 

a  57  and  123  thousandths,  181  and  28  hundredths,  49  and 
3  tenths. 

6  167  and  4  tenths,  2128  and  4  hundredths,  396  and  4 
thousandths. 


28  INTEGERS  AND   DECIMALS 

c  821  and  47  hundredths,  526  and  47  thousandths,  2936 
and  1  tenth. 

d  674  and    37   hundredths,    25824   and   128   thousandths, 
1948  and  4  tenths. 

e  666,666    and  6    tenths,    7,777,777   and    77   hundredths, 
88,888,888  and  888  thousandths. 

201.  Add: 

a  7.3  in.  and  4.7  in. 

b   8.4  sq.  yd.  and  5.6  sq.  yd. 

c    3.35  sq.  ft.  and  4.95  sq.  ft. 

202.  .1  is  how  many  times  as  great  as  .01  ? 

203.  In  the  expression  333.333,  which  3  expresses  the  great- 
est value  ?     The  least  value  ? 

204.  3  in  the  first  integral  place  expresses  how  many  times 
as  much  as  3  in  the  first  decimal  place  ? 

205.  5  in  the  first  decimal  place  expresses  how  many  times 
as  much  as  5  in  the  third  decimal  place  ? 

206.  Which  figures  in  a  decimal  stand  for  the  greater  value, 
those  near  the  decimal  point  or  those  far  away  from  it  ? 

'•  I 

TJ  "O 


w       ^  fl        T3  13         3         3       73 

1  .  1  f    I  '  I    a.   *-  '  :    .1   I    I    I    I-  I 

1  I  |  a  I  I  I  :  HS  1  a  I  I  1 

207.  GLASS  EXERCISE.     Copy  the  diagram,  placing  a  figure 
in  each  decimal  place,  and  read  the  numbers  thus  expressed. 

208.  How  much  is  TJ¥  of  .01  ?     TV  of  .001  ?     ^  of  .0001  ? 
of  .00001  ? 


ADDITION   OF  DECIMALS  29 

209.  Give  values  of  x: 

a  x  =  1  in  the  4th  decimal  place. 
b  x  =  1  in  the  6th  decimal  place.  ^ 
c  x  =  4  in  the  5th  decimal  place. 
d  x  =  7  in  the  4th  decimal  place. 
e  q  =  6  in  the  6th  integral  place. 
/  x  =  2  in  the  6th  decimal  place. 

210.  Add,  and  write  the  sum  in  words : 

a  b  c 

1.235  24375  325685 

123.5  2.4375  .325685 

.1235  .24375  3.25685 

211.  Write  and  read  a  decimal  of  4  places.     Of  5  places. 
Of  6  places. 

212.  Which  integral  place  is  occupied  by  millions  ?     Which 
decimal  place  by  millionths  ? 

213.  Give  the  places  of  the  following:   Thousands,  thou- 
sandths, ten-thousands,  ten-thousandths,  hundreds,  hundredths, 
hundred-thousands,  hundred- thousandths. 

214.  The  expression  .3  shows  that  some  unit  is  divided  into 
10  equal  parts,  and  that  3  of  those  parts  are  taken.     What 
does  .003  show  ?     .0003  ?     .00003  ?    .000003  ? 

215.  When  we  have  .3  of  an  inch,  what  unit  has  been 
divided  into  10  equal  parts  ?    Explain  the  expression  "  .7  of 
a  foot."     « .17  of  a  dollar." 

216.    If  A  and  C  were  each  3  in.  from  It, 

1 ' '    how  far  apart  would  they  be  ? 


217.    If  A  and  Owere  each  122.57  mi.  from  B,  how  far  apart 
would  they  be  ? 

218.    If  A  is  7.7  in.  from  B,  and  C  is 
"J £ ?   3.8  in.  from  B,  how  far  apart  are  J^and  <7? 


UNIVERSITY 


30  INTEGERS  AND   DECIMALS 

219.  CLASS  EXERCISE.      Let  pupils  give  different  lengths 
(with  decimals)  to  AB  and  BC,  and  find  distance  from  A  to  (7. 

220.  By  rail  the  distance  from  Nashville,  Tenn.,  to  Evans- 
ville, Ind.,  is  155.07  mi.,  and  from  Evansville  to  Chicago,  111., 
287.15  mi.     Mary  Allen  lives  in  Nashville.     How  far  will  she 
travel  in  going  from  her  home  through  Evansville  to  Chicago 
and  returning  by  the  same  route  ? 

221.  How  many  cents  equal  1  hundredth  of  a  dollar  ?     .17  ? 

222.  Add: 


a 

b 

c 

d 

$48.33 

$  75.25 

$  81.39 

$  813.45 

76.48 

38.60 

47.50 

425.15 

13.15 

49.76 

86.72 

327.40 

SUGGESTION  TO  TEACHER.  Call  the  attention  of  pupils  to  the  fact  that 
they  have  been  using  decimals  of  a  dollar  in  their  work  with  dollars, 
dimes,  and  cents. 

SUBTRACTION    OF    DECIMALS 

223.  Subtract: 

a  b  c  d  e 

446.35         674.37         821.42         123,478.008  964,821.88 

29.78          338.49          365.17  1,939.981  283,464.79 

224.  Which  is  greater  and  how  much,  .6  of  a  dollar  or 


225.  Which  is  greater,  .1  or  .10?     .4  or  .40?     .50  or  .5? 
.7  or  .700? 

226.  Write  a  decimal.     Annex  a  cipher  to  it,  and  tell  how 
the  value  of  the  decimal  is  affected. 

Without  changing  values, 

227.  Change  to  hundredths  :  .7.     2.1.    45.3. 

228.  Change  to  thousandths  :  .25.     .4.     8.1.     2.56. 

229.  Change  to  ten-thousandths  :  .125.     2.4.     .17. 


SUBTRACTION  OF   DECIMALS  31 

230.  Change  to  hundred-thousandths :  .4758.  3.56.  .417.  .9. 

231.  Change  to  millionths:  .85674.  18.35.  42.7.  .489.  .9249. 

232.  Write  the  expression  .3.     Place  a  cipher  between  the 
decimal  point  an<J  the  figure  3.     How  is  the  value  of  the 
expression  .3  changed  by  placing  the  cipher? 

233.  .5  of  a  dollar  equals  how  many  cents  ?     .05  of  a  dollar 
equals  how  many  cents  ?     Find  their  difference. 

abed 

234.  From          175.5  691.15  436.4  827.3 
take             20.35          420.615          125.25          121.125 

235.  Mr.  Adams  had  75.1  acres  of  land  and  sold  13.4  acres. 
How  many  acres  had  he  left  ? 

236.  From  195.35  sq.  yd.  take  37.15  sq.  yd. 

237.  Find  the  values  of : 

a       1  -  .04  c         2  - 1.8  e     800  -  .390 

b  300  -  .08  d  6001  -  40.683         /  8602  -  304.407 

238.  Write  decimally  11  tenths.     4125  thousandths. 

239.  11  -  11  tenths  =  ?     113  -  113  tenths  =  ? 

240.  117  —  134    thousandths  =  ?       8    tenths  —  436    thou- 
sandths =  ? 

241.  297  -  4138  thousandths  =  ?    480  thousand  -  483  thou- 
sandths =  ? 

242.  From  one  take  three  hundred  seventy-one  thousandths. 

243.  From  two  and  three  hundred  forty-seven  thousandths 
take  eight  hundredths. 

244.  From  ten  thousand  take  ten  thousandths. 

245.  From  ten  millions  take  ten  hundredths. 

246.  From  five  hundred  take  five  hundredths. 


32  INTEGERS  AND  DECIMALS 

247.  From  eight  hundred  thousand  take  eight  thousandths. 

248.  From  five  tenths  take  five  hundredths. 

249.  /Use  12.75  as  a  minuend  with  3.50  as  a  subtrahend  and 
read  the  difference. 

250.  A  merchant  bought  goods  for  $  89.35  and  sold  them 
for  $  125.75.     How  much  did  he  gain  ? 

How  much  is  gained  on  goods : 

251.  Bought  for  $  129.37,  sold  for  $  178.12  ? 

252.  Bought  for  $  363.48,  sold  for  $  429.95  ? 

253.  Bought  for  $  428.35,  sold  for  $  516.81  ? 

254.  Bought  for  $  596.47,  sold  for  $  731.97  ? 

255.  Bought  for  $  1028.50,  sold  for  $  1296.75  ? 

256.  Bought  for  $  1534.81,  sold  for  $  2346.55  ? 

257.  Bought  for  $  .17,  sold  for  $  .23  ? 

How  much  is  lost  on  goods : 

258.  Bought  for  $  275.37,  sold  for  $  179.33  ? 

259.  Bought  for  $  186.38,  sold  for  $  175.47  ? 

260.  Make  problems  about  buying  and  selling. 

261.  Add  875.15  to  itself. 

262.  Add  324.75  to  the  number  that  is  4  more  than  324.75. 
Add  324.75  to  the  number  that  is  .2  more  than  324.75.     Add 
324.75  to  the  number  that  is  .25  more  than  324.75. 

263.    Draw   the    line    XZ   4    in.    long. 

x       T"      z    Mark  the  point  Y,  1  in.  from  Z.      How 

long  is  the  line  XY? 

264.    If  XZ  were  8.8  inches  and  YZ  were  2.2  inches,  how 
long  would  the  line  XFbe  ? 


SUBTRACTION   OF  DECIMALS  33 

265.  If   XZ  were   10   in.,  and   XY  were  7  in.,  how  long 
would  YZ  be  ? 

266.  If  XY  were  t.8  in.,  and  XZ  were  10.9  in.,  how  long 
would  YZ  be  ? 

267.  Harold  stands  7.8  rd.  directly  east  of  a  certain  point, 
and  his  brother  Stanley  stands  15.6  rd.  directly  west  of  it. 
How  far  apart  are  the  boys  ?     Represent  by  lines. 

268.  Stanley    and    Harold    measured    lines    on    the "  floor. 
Harold  started  in  a  corner  and  measured  3  ft.  along  by  the 
side  wall.     Stanley  measured  5  ft.  from  the  same  corner  in 
the  same  direction.     How  far  apart  were   the   ends  of   their 
lines  ? 

SUGGESTION  TO  TEACHER.  Let  two  boys  take  the  parts  of  Harold  and 
Stanley  for  the  benefit  of  those  who  cannot  imagine  the  conditions. 

269.  In  problem  268  if  Harold's  line  were  7.6  ft.  long  and 
Stanley's  9.8  ft.  long,  what  would  be  the  distance  between  the 
ends  of 'their  lines  ? 

270.  Stanley  was  at  one  end  of  a  side  wall  of  a  room  21  ft. 
long,  and  Harold  was  at  the  other  end.     Stanley  walked  7  ft. 
in  the  direction  of  Harold,  and  Harold  walked  2  ft.  toward 
Stanley.     How  far  apart  were  they  then  ? 

SUGGESTION  TO  TEACHER.  For  a  class  exercise  let  children  find  length 
or  width  of  the  schoolroom,  imagine  or  enact  movements  like  those  in  the 
previous  problems,  and  find  distances. 

271.  A  room  is  30  ft.  long.     If  Stanley  walked  from  one 
corner  of  it  7.1  ft.  toward  Harold  who  is  at  the  other  end  of 
the  same  side  wall,  and  Harold  walked  7.9  ft.  toward  him,  how 
far  apart  would  they  be  ? 

272.  High-water  mark  at  a  certain  town  on  the  Ohio  Elver 
was  38.3  ft.  one  year  and  42.1  the  next  year.     How  much 
higher  did  the  river  rise  the  second  year  than  the  first  ? 

HORN.    GRAM.    SCH.    AR. 3 


34 


INTEGERS   AND   DECIMALS 


273.  Find  the  cost  of   the  materials   for  a  Thanksgiving 
dinner  at  the  house  of   Mr.  Smith.     Turkey  $  1.75 ;    oysters 
$  .55 ;    potatoes    $  .05  ;    other  vegetables   $  .10 ;    bread  $  .05  ; 
pickles  $.10;   jelly  $.10;   plum  pudding  $.40;   mince   pies 
$.20;  milk  $.10;  coffee  $.05;  salt,  pepper,  sugar  (estimated) 
$.05;   nuts  and  raisins  $.30.     The  family  consisted  of  Mr. 
and   Mrs.   Smith   and  six  children.     What  was  the  average 
cost  for  each  person  ? 

274.  Plan  an  ordinary  dinner  and  its  cost. 

NOTE  TO  TEACHER.  For  the  following 
work  pupils  must  be  provided  with  rulers 
showing  the  decimeter,  centimeter,  and 
millimeter. 

275.  How  many  centimeters  in  3 
decimeters  ?     In  5^  decimeters  ? 

276.  How  many  millimeters  in  5 
centimeters  ?     In  7|  centimeters  ? 

277.  How  many  millimeters  in  a 
decimeter  ? 

278.  Draw  on  the  board  a  line  10 
dm.  long.     Its  length  is  1  meter.     A 
decimeter    equals    what    part    of    a 
meter?     A   centimeter   equals  what 
part  of  a  meter  ? 

279.  The   Latin   word   "centum" 
means  100.     How  many  cents  make 
a   dollar  ?     How   many   centimeters 
make  a  meter  ? 

SUGGESTION  TO  TEACHER.  Let  some 
pupils  make  meter  sticks,  marking  the  sub- 
divisions of  1  dm.  and  of  1  cm.  Let  others 
mark  off  a  meter  and  its  subdivisions  on 
ribbon  or  tape.  Keep  the  best  of  these  a's 
a  part  of  the  school  apparatus. 


SUBTRACTION   OF   DECIMALS  35 

280.  What'is  meant  by  the  perimeter  of  a  figure  ? 
Draw  a  square  centimeter.  How  many  centimeters  in 
its  perimeter  ?  How  many  millimeters  ? 

281.  Draw  a  square  decimeter.     How  many  decimeters  in  its 
perimeter  ?    How  many  centimeters  ?    How  many  millimeters  ? 

282.  Draw  a  line  on  the  board  1  decimeter  and  6  centimeters 
long.      Lengthen    it    4   centimeters.      How  many  decimeters 
long  is  it  now  ?     How  many  centimeters  ? 

283.  A  millimeter  is  what  part  of  a  centimeter  ?    Of  a  deci- 
meter ?     Of  a  meter  ? 

284.  M.  stands  for  meter;  dm.  for  decimeter;  cm.  for  centi- 
meter, and  mm.  for  millimeter.     Can  you  see  why  ? 

SUGGESTION  TO  TEACHER.  Let  pupils  find  in  metric  measurements 
the  length  of  room,  book,  desk,  writing  tablet,  pencil,  penholder,  door, 
blackboard,  or  any  other  object. 

285.  Draw  a  line  1.3  cm.  long.     How  many  millimeters  long 
is  it? 

286.  How  many  millimeters  in  3  cm.  ?     In  7  cm.  ?     9  cm. 
and  4  mm.  ?     2  dm.  ?     3  dm.  and  4  cm.  ?     5  dm.  and  2  cm.  ? 

The  metric  system  is  a  very  convenient  way  of  measuring,  because  a 
unit  of  each  denomination  is  ^  of  a  unit  of  the  next  higher.  As  it  is 
used  in  government  service,  every  child  should  learn  it. 

287.  Compare  1  cm.  with  1  in.     1  dm.  with  4  in.     1  m.  with 
lyd. 

288.  Draw  the  rectangle  ABCD,  making  the  base  8.4  cm. 
A B    and  the  altitude  5.6  cm.     How  long  is  the 

perimeter  ?     Describe  a  rectangle. 

289.  Draw  in  your  rectangle  the  line 
AC.  A  line  drawn  from  one  angle  of  a 
figure  to  another  angle  that  is  not  next  to 


FIG.  1.  it  is  called  a  Diagonal.     Draw  as  many 

diagonals  as  you  can  in  your  rectangle  ABCD.     Which  diago- 
nal is  the  longer  ? 


36 


INTEGERS  AND   DECIMALS 


Can  you  draw  a  diagonal  of  the 


290.  Draw  a  triangle, 
triangle  ?     Explain. 

291.  Two  boys  were  in  diagonally  opposite  corners  of  a 
room.     The  length  of  the  diagonal  was  35  ft.     If  each  walked 
3.5  ft.  toward  the  other,  how  far  apart  would  they  be  ? 

SUGGESTION  TO  TEACHER.  Let  children  find  the  length  of  the  diago- 
nal of  the  floor.  Two  pupils  may  stretch  a  string  from  opposite  corners 
at  a  convenient  height  parallel  to  the  floor,  and  hence  perpendicular  to  the 
intersection  of  the  side  walls,  and  then  measure  the  string.  Use  the  terms 
"diagonal,"  "parallel,"  and  "perpendicular,"  and  let  children  measure 
and  adjust  until  parallelism  is  secured.  Let  pupils  give  to  the  class  prob- 
lems similar  to  Ex.  291. 

292.  By  measuring,  find  the  length  of  the  diagonals  of  the 
cover  of  your  arithmetic.     Of  the  top  of  your  desk. 

8.4  4.2  293.    How  long  is  the  perimeter  of 

Fig.  2  if  the  dimensions  are  centimeters  ? 

NOTE  TO  TEACHER.  The  expression  "  Fig.  2  " 
is  used  for  the  sake  of  brevity  instead  of  the 
more  exact  expression,  "  The  figure  repre- 
sented by  Fig.  2."  This  contraction  is  used 
throughout  the  book. 

294.  How  long  is  the  perimeter  of 
the  triangle  ABC  ?     The  measurements 
are  given  in  centimeters. 

295.  Triangles   that  have  two  sides 
equal    are    called    Isosceles     Triangles. 
Draw   two   lines    of    the    same    length 
meeting  at  a  point.     Join  the  ends  of 
the   lines   by   a    straight    line.      What 

FIG.  3.  kind  of  a  triangle  have  you  drawn  ? 

296.  Draw  an  isosceles  triangle  whose  equal  sides  are  each 
7  cm.  long. 

297.  The  side  of  a  triangle  upon  which  it  is  supposed  to 
stand  is  called  its  Base.     One  of  the  equal  sides  of  Fig.  3  is 
how  much  longer  than  its  base  ? 


n 

TjJ 

<« 

•** 

4.2 

16.8 
FIG.  2. 

SUBTRACTION  OF   DECIMALS 


37 


10.2  cm. 
FIG.  4. 


1.08  cm. 
Fia.  6. 


298.  What  is  the  sum  of  the  equal  sides 
of  Fig.  3  ?     The  sum  of  the  equal  sides  is 
how  much  more  than  the  base  ? 

299.  How  long  is  the  perimeter  of  Fig.  4  ? 
The  sum  of  the  equal  sides  is  how  much 
more  than  the  base  ? 

300.  How  long  is  the  perimeter  of  Fig.  5  ? 
What  kind  of  a  triangle  is  it  ?     Why  ? 

301 .  How  long  is  the  perimeter  of  Fig.  6  ? 
Is  it  an  isosceles  triangle  ?    Explain.    The 
sum  of  the  two  shorter  sides  is  how  much 
more  than  the  longest  side  ? 

302.  The  side  AB  is  how  much  longer 
than  the  side  AC?     Than  the  side  BC? 
The  sum  of  AB  and  AC  is  how  much  more 
than  BC?     The  sum  of  AB  and  BC  is 
how  much  more  than  AC?     The  sum  of 

B  AC   and    BC   is    how  much    more    than 
AB? 


303.  Name  the  denominations  in  order  from  a  millimeter  to 
a  meter. 

304.  Express  in  decimal  form  the  part  which  one  unit  of 
each  lower  denomination  is  of  one  meter. 

305.  In  the  perimeter  of   a  square  decimeter,  how  many 
centimeters  ?     How  many  millimeters  ? 

306.  If  there  are  5  sq.  cm.  in  a  row,  how  many  square  centi- 
meters are  there  in  a  rectangle  composed  of  3  rows  ?     7  rows  ? 

307.  If  there  were  5  sq.  cm.  in  a  row,  how  many  rows  would 
it  take  to  make  a  perfect  square  ? 

308.  If  there  were  5  sq.  cm.  in  a  row,  how  many  rows 
would  it  take  to  make  a  rectangle  containing  30  sq.  cm.  ? 
40  sq.  cm.  ? 


38  INTEGERS  AND   DECIMALS 

309.  How  many  rows  would  it  take  to  make  a  perfect  square 
if  in  each  row  there  were  3  sq.  cm.  ?     4  sq.  cm.  ?     6  sq.  cm.  ? 

310.  How  many  square  millimeters  in  a  square  centimeter? 

311.  Draw  a  square  decimeter.     How  many  square  centi- 
meters in  it  ?     How  many  square  millimeters  ? 

312.  Draw  on  the  floor  a  square  meter  and  divide  it  off  into 
square  decimeters.     Divide  one  of  the  square  decimeters  into 
square  centimeters.     Can  you  easily  divide  square  centimeters 
on  the  floor  into  square  millimeters  ?     Explain. 

313.  In  a  square  meter,  how  many  square  decimeters  ?    How 
many  square  centimeters  ?     How  many  square  millimeters  ? 

314.  Express  in  decimal  form  the  part  which  one  unit  of 
each  lower  denomination  is  of  one  square  meter. 

315.  John  had  10  cents.     He  spent  6  cents  for  a  ball  and 
3  cents  for  a  top.     How  much  had  he  left  ?     His  father  had 
$  537.84.     He  bought  a  horse  and  a  carriage  for  $  300.  and  a 
set  of  harness  for  $  19.75.     How  much  had  he  left  ? 

316.  One   day  a  bank   cashier  paid  out   seven   thousand- 
dollar  bills.     On  the  next  he  paid  out  seven  hundred-dollar 
bills.     How  much  more  did  he  pay  out  on  the  first  day  than 
on  the  second  ? 

317.  A  coat  that  cost  $9.75  was  sold  for  $12.50.     How 
much  was  gained  ? 

318.  A  coat  that  cost  $14.75  was  sold  for  $13.50.     How 
much  was  lost  ? 

319.  A  ball  that  cost  6  cents  was  sold  so  as  to  gain  1  cent. 
For  how  much  was  it  sold  ?     A  horse  that  cost  $  115  was  sold 
so  as  to  gain  $  17.35.     For  how  much  was  it  sold  ? 

320.  By  selling  a   horse   for   $475.50,  Mr.   Smith  gained 
$87.75.      How   much   did    the   horse    cost?      Make    similar 
problems. 


MULTIPLICATION  OF   DECIMALS  39 

321.  Mr.  Cox  spent  $237.38  in  May,  and  $348.31  in  June. 
How  much  more  did  he  spend  in  June  than  in  May  ? 

322.  Mr.  Ward  deposited  $  89.25  in  the  bank  on  Monday, 
and  on  Tuesday  $  48.55.    On  Wednesday  he  drew  out  $  105.35. 
How  much  remained  to  his  credit  in  the  bank  ? 

323.  Have  you  ever  seen  a  bank  ?     If  so,  describe  it. 

324.  I  had  two  notes  due  me,  one  of  $  420  and  another  of 
$  266.66.     How  much  was  still  due  me  after  $  389.50  was  paid  ? 

SUGGESTION  TO  TEACHER.     Show  promissory  note  and  explain  its  use. 

325.  Mr.  Gage  bought  a  piece  of  land  of  Mr.  Wood  and 
gave  him  his  note  for  $  700*    When  the  interest  on  the  note 
amounted  to  $  38.?5,  Mr.  Gage  paid  $  500  on  it.     How  much 
did  he  still  owe  ? 

326.  When  $56.25  interest  was  due  on  the  amount  Mr. 
Gage  then  owed  he  paid  $  175.25.    How  much  did  he  still  owe  ? 

MULTIPLICATION  OF  DECIMALS 

327.  Multiply  1.2  by  3. 

When  2  tenths  are  multiplied  by  3,  the  result  is  6 
tenths,  just  as  2  units  multiplied  by  3  are  6  units,  or 
T^  as  3  times  2  oranges  equal  6  oranges. 

328.  When  a  decimal  is  multiplied  by  a  whole  number, 
there  are  as  many  decimal  places  in  the  product  as  there  are 
in  the  multiplicand.     Multiply  4.75  by  5.     By  20.     By  30. 

329.  Give   rapidly  the   products  obtained  by  multiplying 
each  of  the  following  numbers  in  succession  by  each  integer 
between  1  and  13  :     .6.     .8.     .9.     1.2.     .12.     .012. 


a 

b 

c 

d 

e 

330. 

Multiply 

478.37 

21.175 

9.35 

2.3 

24.7 

by 

6 

11 

36 

24 

81 

331.    What  is  the  cost  of  a  dozen  hats  at  $  3.75  apiece  ? 


40 


INTEGERS   AND   DECIMALS 


FIG.  7. 


332.  Multiply  1.28415  by  the  third  even  number. 

333.  All  numbers  that  are  not  even  are  called  Odd  Numbers. 
Write  in  order  the  first  eight  odd  numbers  and  find  their  sum. 

334.  Every  odd  number  ends  with  one  of  5  digits.     Name 
them. 

335.  How  many  of  the  first  19  numbers  are  odd  ? 

336.  Multiply  8.8571  by  the  fourth  odd  number. 

337.  Multiply  16.754  by  the  seventh  odd  number. 

338.  An  Equilateral  Triangle  is  a  tri- 
angle all  of   whose  sides   are   equal,  as 
Fig.  7.     If  each  side  of  Fig*  7  were  8.75 
in.  long,  how  long  would  its  perimeter  be  ? 

SUGGESTION  TO  TEACHER.  Show  the  follow- 
ing method  of  constructing  an  equilateral  tri- 
angle. Draw  a  line  of  convenient  length  for 
base  BC  as  in  Fig.  8.  With  B  as  a  center  and 
BC  as  a  radius,  draw  an  arc.  With  C  as  a 
center  and  CB  as  a  radius,  draw  a  second  arc 
intersecting  the  first  at  A.  Draw  A  B  and  A  G. 
Erase  construction  lines. 

339.  Construct  an  equilateral  triangle 
each  side  of  which  is  4  in.     How  long 
is  its  perimeter  ?     How  long  would  the 
perimeter  be  if  each  side  were  8.25  in.  ? 
4.875  in.  ? 

340.  In  the  triangle  ABC  each  of  the  equal 
sides  is  twice  as  long  as  the  base.     What  kind  of 
a  triangle  is  it  ?     How  long  is  the  perimeter  ? 

341.  Construct   an  isosceles   triangle   each  of 
whose  equal  sides  is  twice  the  base. 

SUGGESTION  TO  TEACHER.  Let  pupils  use  the  same 
method  of  construction  as  is  used  for  equilateral  triangles 
except  that  each  arc  should  be  drawn  with  a  radius  twice 
as  long  as  the  base. 


FIG. 


4.79 


FIG.  9. 


MULTIPLICATION   OF   DECIMALS  41 

342.  In  the  triangle  ABC,  AB  and  AC  are 
each  3  times  as  long  as  BC.     How  long  is  the 
perimeter  ?     The  sum  of  the  equal  sides  is  how 
much  more  than  the  base  ? 

343.  Construct  an  isosceles   triangle   each    of 
whose  equal  sides  is  3  times  as  long  as  the  base. 
The  perimeter  is  how  many  times  as  long  as  the 
base  ? 

344.  Construct  an  isosceles  triangle  whose  base 
J  -  :  -  J,     is  4.5  inches  and  each  of  whose  equal  sides  is 

Fia.  10.        6.5  in.     The  perimeter  is  how  much  longer  than 
the  base  ? 

345.  Construct  an  isosceles  triangle  whose  base  is  3  in.  and 
the  sum  of  whose  equal  sides  is  9  in. 

346.  How  much  will  a  dozen  knives  cost  at  $  .87  apiece  ? 

347.  John  had  $  .15  which  was  i  of  what  he  needed  to  buy 
a  music  book.     What  was  the  price  of  the  book  ? 

348.  .00256  is  |  of  what  number  ?     ^  of  what  ? 

349.  If  a  dozen  knives  are  bought  at  $  .67  apiece  and  sold 
for  $  1.00  apiece,  how  much  is  gained  ? 

350.  The  rent  of  a  house  is  $  17.50  per  month.     How  much 
is  the  rent  for  a  year  ? 

351.  A  man  spends  on  the  average  $  .25  a  day  for  cigars. 
How  much  does  he  spend  in  a  leap  year  ? 

NEW  YORK,  Sept.  12,  1898. 

352.  MR.  WM.  H.  MORSE 

Bought  of  THOMAS  D.  LONG, 

25  Ib.  Sugar    .....     @  $  .05     ....  $  1.25 

17  Ib.  Coffee    .....     @      .25     ....  4.25 

6  Ib.  Tea  ......              .87     ....  5.25 


$  10.75 
Received  Payment, 

THOMAS  D.  LONG. 


42  INTEGERS  AND  DECIMALS 

353.  Make  a  bill  similar  to  Ex.  352,  in  which  the  price  of 
the  sugar  is  6^  per  lb.,  the  coffee  30^,  and  the  tea  95^. 

354.  Imagine  that  you  are  a  clerk  in  a  store  where  a  cus- 
tomer buys  the  following  bill  of  goods.     Make  out  the  bill  and 
receipt  it. 

8  yd.  Gingham @  $  .371  per  yd. 

9  yd.  Binding @      .07    per  yd. 

11  yd.  Percale @      .11    per  yd. 

2         Fans @      .75    each. 

SUGGESTION  TO  TEACHER.  Get  bill  heads  from  merchants  and  let 
them  be  copied  in  the  following  exercises. 

355.  Imagine  yourself   to  be  a  clerk   in  a   grocery  store. 
Make  out  and  receipt  a  bill  of  goods  bought  by  Mr.  James 
Jones. 

356.  Make  out  and  receipt  a  bill  of  goods  bought  in  (a)  a 
dry  goods  store.    (6)  A  shoe  store,     (c)  A  music  store,     (d)  A 
toy  store,     (e)  A  clothing  store. 

357.  Make  out  a  meat  bill  for  Mr.  Walter  Smith  for  the 
week  ending  Saturday,  Sept.  8,  1900. 

358.  When  4.8  is  changed  to  48,  by  what  is  it  multiplied  ? 
By  what  must  4.8  be  multiplied  to  make  it  480  ? 

SUGGESTION  TO  TEACHER.  Show  the  method  of  multiplying  by  any 
power  of  10  by  moving  the  decimal  point  to  the  right,  and  of  dividing  by 
any  power  of  10  by  moving  the  decimal  point  to  the  left. 

359.  Multiply  1.357  by  10.     By  10000.     By  100000. 

360.  Divide  125.7  by  10.     By  100.     By  1000.     By  100000. 

361.  Add  1.25  to  100  times  itself. 

362.  Add  to  3.25  the  number  that  is  y^  of  it. 

363.  Subtract  from  875  the  number  that  is  -fa  of  it. 

364.  Subtract  .213  from  1000  times  itself. 

365.  3.78  is  how  much  less  than  1000  times  itself? 


MULTIPLICATION   OF  DECIMALS  43 

366.  How  do  you  multiply  an  integer  or  a  decimal  by  10  ? 
By  100  ?     By  1000  ?     By  any  number  expressed  by  1  with  one 
or  more  ciphers  annexed  ? 

367.  How  is  an  integer  or  a  decimal  divided  by  10?     By 
1000?     By  any  number  expressed   by  1  with   one  or   more 
ciphers  annexed? 

368.  How  much  is  .1  or       of  30?     .2  of  30? 


369.  How  much  is  .01  or  T^¥  of  300  ?     .02  of  300  ? 

370.  How  much  is  .01  of  375  ?     .02  of  375  ? 

371.  Write  an  integer  of  three  places  and  find  .03  of  it. 

372.  Write  an  integer  of  four  places  and  find  .7  of  it. 

373.  When  an  integer  is  multiplied  by  a  decimal  there  are 
as  many  decimal  places  in  the  product  as  there  are  in  the  mul- 
tiplier.    Multiply  325  by  .7.     By  .13.     By  .125. 

a  b  c  d  e 

374.  Multiply         275          283  413          671          1289 

\A          ^17          1.01  _.21  .001 

375.  How  much  is  .25  of  a  square  8  in.  in  dimensions? 

376.  With  375  as  a  multiplicand  and  .31  as  a  multiplier, 
what  is  the  product  ? 

377.  With  145  as  a  multiplicand  and  .41  as  a  multiplier, 
what  is  the  product  ? 

378.  How  many  places  and  in  which  direction  must  the 
decimal  point  be  moved  in  order  to  divide  125.7  by  100,  or  to 
find  y-^  of  it? 

379.  Find  .01  of  217.25.     Of  365.7.     Of  412.137. 

380.  How  much  is  .01  of  225.7?     .03  of  it?     .08  of  it? 
Compare  the  number  of  decimal  places  in  the  products  with 
the  number  of  decimal  places  in  multiplicand  and  multiplier. 

381.  When  an  integer  is  multiplied  by  a  decimal,  how  must 
the  product  be  pointed  off  ? 


44  INTEGERS  AND  DECIMALS 

382.  How  must  the  products  be  pointed  off  when  a  decimal 
is  multiplied  by  an  integer  ? 

383.  When  a  decimal  is  multiplied  by  a  decimal,  the  product 
contains  as  many  decimal  places  as  there  are  decimal  places  in 
both  multiplicand  and  multiplier.     Multiply  .05  by  .5. 

384.  Find  products : 

a  1.57  x  .3  /  84.2  x  .43 

b  14.5  x  .7  g  1.32  x  4.1 

c  41.42  x  6  h  6.71  x  .11 

d  2.42  x  1.21  t  3.41  x  .701 

e  3.43  x  6.41  j  1.2  x  .41 

385.  If  there  are  not  as  many  figures  in  the  product  as 
there  are  decimal  places  in  both  multiplicand  and  multiplier, 
ciphers  must  be  prefixed  to  the  product  before  pointing  it  off. 
Explain. 

386.  Multiply  .15  by  .3.    .35  x  .07.    .002  x  .7.    .021  x  .008. 

387.  To  square  a  number  is  to  multiply  it  by  itself.     Square : 
1.5.    .16.    2.3.     .009.    .18.    1.9.    3.2.    .051.    2.8.    4.08.    .025. 

388.  Draw  a  square  whose  dimensions  are  1.5  dm.      How 
many  square  decimeters  in  it  ? 

389.  How  many  square  decimeters  in  a  rectangle  1.3  dm. 
long  and  1.2  dm.  wide  ?     Represent. 

390.  How  many  square  feet  in  a  square  whose  dimensions 
are  1.25  in.  ?     Eepresent. 

391.  How  many  square  inches  in  a  rectangle  7.5  in.  long 
and  3.5  in.  wide  ? 

392.  A  lot  cost  $  687.50,  and  the  house  which  stood  upon  it 
cost  4.5  times  as  much.    How  much  did  the  house  cost?    How 
much  did  both  cost  ? 


MULTIPLICATION   OF   DECIMALS 


45 


393.  Kate  drew  a  rectangle  4.5  in.  long  and  2.75  in.  wide. 
Anna  drew  a  rectangle  3.5  times  as  large  as  Kate's  rectangle. 
How  many  square  inches  in  Anna's  rectangle  ? 

394.  One  hundredth  of  anything  is  called  1  per  cent  of  it. 
Per  cent  is  written  %,  as  4%  means  .04.     Write  as  per  cent : 
.17.     .07.     .03f     ^     .50.     ^      .61. 

395.  What  per  cent  of 
Pig.  11  is  shaded  ?     Un- 
shaded ? 

396.  BOARD    WORK. 

Draw  a  square  decimeter 
and  mark  it  off  into  square 
centimeters.  Shade  13% 
of  it.  What  per  cent  of 
it  is  unshaded  ? 

397.  Shade    ^    of     it. 
What    per    cent    is    un- 

FIG.  11.  Shaded? 

398.  Shade  the  following  parts  of  the  figure  and  tell  in  each 
case  what  per  cent  is  unshaded : 

\        .3          39%          .4          i          57%  .6 

.7         |          78%  f         .9          97%  100% 

399.  What   per   cent   of   a   dollar   is   1   cent?      3   cents? 
21  cents? 

400.  George  had  a  dollar  and  lost  5%  of  it.     How  many 
cents  had  he  left  ?     How  many  cents  had  he  left  when  he  had 
spent  another  5%  of  the  dollar? 

401.  Of  100  words  that  John  wrote  in  a  spelling  test,  13 
were  wrong.     What  was  his  per  cent  on  that  test  ? 


46  INTEGERS  AND  DECIMALS 

402.  When  your   record  in  an  examination  is  99%,  how 
many  himdredths  of  your  work   are   correct  ?      How  many 
hundredths  are  wrong? 

403.  CLASS  EXERCISE.     -  may  name  a  number  less  than 
100,  and  the  class  may  tell  what  per  cent  it  is  of  100,  and  how 
many  per  cent  of  100  it  lacks  of  being  100. 

404.  As  6%  of  anything  is  .06  of  it,  we  may  find  6%  of  any 
number  by  multiplying  it  by  .06.     Find  6%  of  44. 

405.  In  the  same  way  find  6%  of:   28.    39.    63.    144.    135. 

406.  How  would  you  find  1%  or  any  other  per  cent  of  a 
number  ?     Illustrate. 

407.  A  company  of  soldiers  consisted  of  100  men.     7%  of 
them  were  mustered  out.     How  many  soldiers  remained  ? 

408.  Find  5%  of:   14.     24.     75.     1.83.     6.44.     3.72.     8.49. 

409.  Mr.  Miller  bought  $960  worth  of  goods,  and,  in  selling 
them,  gained  15%.     How  much  did  he  gain?     How  much  did 
he  receive  for  them  ? 

410.  Mr.   Low  bought  $3125.50  worth  of  goods  and  sold 
them  at  a  loss  of  2%.     How  much  did  he  lose  ?     How  much 
did  he  receive  for  them  ? 

411.  Make  problems  about  buying  goods  and  selling  them  at 
a  certain  per  cent  of  gain  or  of  loss. 

412.  How  many  per  cent  of  anything  is  the  whole  of  it  ? 
i-  of  it  ?     i  of  it  ?     f  of  it  ? 

413.  Fill  out  the  following  and  learn: 

The  whole  =  100%  \  =  --  % 


414.    Find  25%  of  64  by  multiplying  it  by  .25.     Find  25% 
of  64  by  taking  \  of  it. 


DIVISION  OF  DECIMALS  47 

415.  Find  the  values  of  each  of  the  following  in  two  ways, 
first  by  multiplying  by  the  decimal  fraction  which  the  per  cent 
equals,  and  then  by  a  common  fraction  which  it  equals.     25% 
of  48.     50%  of  12.     75%  of  8.     75%  of  24.     75%  of  32. 

416.  Find  in  the  shortest  way  how  many  men  equal  50%  of 
14  men.     How  many  bu.  in  75%  of  12  bu.  ? 

417.  How  many  inches  in  50%  of  a  foot?     25%  ?     75%  ? 
How  many  quarts  in  25%  of  a  gallon  ?     How  many  quarts  in 
75%  of  a   peck?      How  many  ounces  in  25%  of  a  pound? 
50%  ?     75%  ? 

DIVISION  OF  DECIMALS 

418.  Divide  .75  into  3  equal  parts.     If  75  cents  are  divided 
equally  among  3  persons,  will  the  "25  "  which  each  receives  be 
25  cents  or  25  other  things  ?     If  .75  of  anything  are  divided 
into  3  equal  parts,  one  of  these  parts  will  be  25  what  ? 

419.  When  a  decimal  is  divided  by  an  integer,  there  are  as 
many  decimal  places  in  the  quotient  as  there  are  in  the  divi- 
dend.    Divide  9.24  by  7. 

In  dividing  a  decimal  by  an  integer  by  short  division  place  the  decimal 
7">0  24    Point  of  the  Quotient  directly  under  that  of  the  dividend  as 
'      9    soon  as  it  is  reached. 

In  this  case  the  quotient  of  9  units  divided  by  7  is  1  unit 
with  a  remainder  of  2  units.  The  decimal  point  should  be  placed  after 
the  1  unit  before  the  division  is  continued. 

420.  By  7  divide:    2.583.     1.0332.     4.1328. 

421.  Find  values  of  x: 

a  b  c  d 

19.64  38.82  5.76  _  21.60 

X  =  X  =  X  = X  —  

4686 

e  '   f  9  * 

x  =  343.7         x  =  13.25         g  =  848.8          x  =  1.989 


48  INTEGERS  AND  DECIMALS 

422.  Find  the  length,  of  one  side  of  an  equilateral  triangle 
whose  perimeter  is  7.5  in. 

423.  By  9  divide:    8.811.        34.569.        672.3.        4712.31 

424.  $  12,384.75  were  divided  among  5  heirs.     How  much 
did  each  receive  ? 

425.  In  one  week  Mr.  A.  earned  $  123.66.     What  were  his 
average  earnings  for  each  working  day  of  the  week  ? 

426.  Find  J  of  .0076. 

4).  0076  Queries.     How  many  tenths  in  \  of  .0  ?     How  many 

.0019          hundredths  in  \  of  .00  ?     How  many  thousandths  in  \ 
of  .007  ?     How  many  ten-thousandths  in  \  of  .0036  ? 

427.  Find  1  of  .008.     Of  .016.     Of  .246. 

428.  By  8  divide  :   .01728.       .002016.       .12102.       .025832. 

429.  Divide  .12  by  9,  carrying  the  division  to  three  places 
of  decimals. 

9).  120  Annexing  a  cipher  to  .12,  we  have  .120,  which  is 

.0131         equal  in  value  to  .12.     .120  divided  by  9  equals  . 


430.  How  many  ten-thousandths  in  the  quotients  of  the 
following  ? 

1.34  87.1  .128  .542  76.4 

69783 

431.  Divide  to  three  places  of  decimals:  H§.   Sfi.   5if§. 

4          o          5 

432.  Divide  22.75  by  13. 
1.75 

13)22.75  In  dividing  a  decimal  by  an  integer  by  long  division, 

13  write  the  quotient  above  the  dividend  and  place  the 

~~97  decimal  point  of  the  quotient  above  the  decimal  point 

?J_  of  the  dividend  as  soon  as  it  is  reached. 
65 

433.  By  21  divide:     8.82.         26.04.         10.353.         4.1349. 

434.  By  32  divide:    5.44.         1.632.         .11424.  20.48. 


DIVISION  OF   DECIMALS  49 

435.  By  24  divide:    .3456.     .5184.     .241584.     .5544. 

436.  Divide  55.44  by  44.     By  28.     By  77. 

437.  If  $  3.15  were   divided   among   15   boys,   how   much 
would  each  receive  ? 

438.  If  hats  are  bought  at  $  8.64  a  dozen,  how  much  does 
one  hat  cost  ? 

439.  If  $  250  were  divided  equally  among  3  men,  how  many 
dollars  and  cents  would  each  man  receive  ? 

440.  If  the  following  sums  of  money  were  divided  equally 
among  five  persons,  how  many  dollars  and  cents  would  each 
person  receive  ? 

$124  $661  $946  $12823  $67847 

441.  What  is  |  of  18.24  ?     Of  17.52?     Of  86.25? 

442.  If  768.32  acres  of   land  were  divided  equally  among 
16  men,  how  many  acres  would  3  men  receive  ? 

443.  If   11   doors  cost   $19.25,  how  much  would  2  doors 
cost? 

444.  Multiply  549.36  by  3£.        By  4J.       By  81        By  12J. 

445.  46.125x21  =  ?  46.125  x3fc=? 

446.  How  much  will  one  knife  cost  at  $  9.00  a  dozen  ?     At 
10.50  per  dozen  ?     At  $  15.00  per  dozen  ? 

447.  How  much  is  gained  on  each  hat  by  buying  hats  at 
$  20  a  dozen,  and  selling  them  at  $2.00  apiece  ? 

448.  How  much  is  gained  on  each  quart  of  milk  : 
a  Bought  at  $  .28  a  gallon,  sold  at  $  .08  a  quart  ? 
b   Bought  at  $  .25  a  gallon,  sold  at  $  .07  a  quart  ? 
c    Bought  at  $  .30  a  gallon,  sold  at  $  .09  a  quart  ? 

449.  In  buying  milk  at  $  .20  a  gallon  and  selling  it  at  $  .06 
a  quart,  how  many  quarts   must  a  milk  dealer   sell   to  gain 
$  1.00  ? 

HORN.    GRAM.    SCH.    AR. 4 


50  INTEGERS  AND   DECIMALS 

450.  In  buying  balls  at  $  1.00  a  dozen  and  selling  them  for 
$  .10  apiece,  how  much  is  gained  on  each  ball  ? 

451.  When  45  yards  of  calico  are  bought  for  $  1.35,  and 
sold  at  $  .05  a  yard,  how  much  is  gained  on  each  yard  ? 

452.  Divide  8.64  by  2.     If  both  dividend  and  divisor  were 
ten  times  as  large  as  they  are,  what  would  the  quotient  be  ? 

SUGGESTION  TO  TEACHER.  The  principle,  "  Multiplying  both  dividend 
and  divisor  by  the  same  number  does  not  change  the  quotient,"  should 
be  thoroughly  reviewed  and  illustrated  before  the  following  work  is  done. 

453.  Divide  3.76  by  .2. 

If  both  dividend  and  divisor  are  multiplied  by  10,  we  have  37.6  ~  2. 
This  is  similar  to  previous  problems. 

454.  By  the  following  rule  perform  this  example  in  division 
of  decimals,  and  give  reason  for  the  rule.     1.96  -r-  .4. 

To  divide  by  a  decimal  — 

Move  the  decimal  point  of  the  divisor  to  the  right  until  the 
divisor  is  an  integer.  Move  the  decimal  point  of  the  dividend 
an  equal  number  of  places  to  the  right,  annexing  ciphers  if 
necessary.  Divide,  and  point  off  as  many  decimal  places  in  the 
quotient  as  there  are  then  in  the  dividend. 

455.  Divide  each  of  the  following  by  .09  : 

1.125  12.33  43.119  62.91  4.815 

456.  Use  .06  as  a  divisor  with  the  following  dividends  : 
221.4         13.2        54.6        91.2         .636        5940        2100 

457.  Find  values  of  x  : 


- 


a 
78.3 

/ 
168 
.35 

b 

x  =  Ws 

9 

x-7S 

X~T5 

c 

*-49-7 

d 
x-37-5 

e 
x-SSA 

'  .14 
k 

X-65A 
"12 

".15 

i 
^_165 

~  .08 

i 

a-89-1 

~^06 

MISCELLANEOUS   EXERCISES  51 

458.  A   music   teacher   earned   $  100  in   a   month,    giving 
lessons  at  $  1.25  each.     How  many  lessons  did  she  give  ? 

459.  At  75^  a  yard,  how  many  yards  of  lace  can  be  bought 
for  $  12.75  ?     For  $  23.25  ? 

460.  Find  quotients : 

Dividend  Divisor  Dividend  Divisor 

a  2.25              1.5  /  2.057  12.1 

b  2.75              2.5  g  3.144  1.31 

c  137.5  1.25  h  539.6  14.2 

d  396               1.2  i  114.92  .221 

e  4.84              1.1  j  603.2  .232 

461.  How  long  is  a  rectangle  which  is 

a  5  ft.  wide,  and  contains  35  sq.  ft.? 
b  .3  ft.  wide,  and  contains  .75  sq.  ft.? 
c    .7  ft.  wide,  and  contains  .77  sq.  ft.? 
d  .5  cm.  long,  and  contains  .125  sq.  cm.? 
e   .9  in.  long,  and  contains  .72  sq.  in.? 

MISCELLANEOUS   EXERCISES 

1.  Add,  1248.375,  115.67241,  3935.5428,  and  138.463249. 

2.  From  13  thousand  and  21  thousandths  take  11  hundred 
and  4  hundredth  s. 

3.  Multiply  .246  by  .89. 

4.  Divide  243.26647  by  .98. 

5.  Write  in  Arabic  notation  and  find  the  sum :     MI,  MV, 
MX,  ML,  MC,  MD. 

6.  Find  the  sum  of  all  the  numbers  less  than  100  that  are 
expressed  in  Eoman  notation  by  2  letters.     By  3  letters.     By 
4  letters.     By  5  letters.     By  6  letters.     By  7  letters. 

7.  Subtract  from   100  the   number  less   than   100  that  is 
expressed  in  Eoman  notation  by  8  letters. 


52  INTEGERS   AND   DECIMALS 

8.  Express  decimally  and  add :    137  and  17  hundredths,  23 
thousand  67  and  19  ten-thousandths,  38  thousand  5  and  11 
millionths. 

9.  From  256  thousand  17  and  15  thousandths  take  128  and 
129  ten-thousandths. 

10.  How  many  square  feet  in  a  rectangle  1.75  ft.  long  and 
1.25  ft.  wide  ?     How  long  is  its  perimeter  ? 

11.  How  wide  is  a  rectangle  that  contains  1.92  sq.  in.  and  is 
1.6  in.  long?     How  long  is  its  perimeter  ? 

12.  1.44  is  how  many  times  .0012  ? 

13.  A  merchant  bought  $2125.75  worth  of  goods,  and  sold 
them  so  as  to  gain  12%  of  the  cost.     How  much  did  he  gain  ? 

14.  Mr.  Duncan  bought  goods  that  cost  him  $  1226.35,  and 
sold  them  so  as  to  gain  16%.     For  how  much  did  he  sell 
them? 

15.  A  man  died,  leaving  $  12,000.     He  willed  50%  of  it  to 
his  wife,  30%  to  his  daughter,  and  the  rest  to  a  library.     How 
much  did  each  receive  ? 

16.  Thomas  bought  a  dime's  worth  of  ice  cream,  which  was 
only  50%  of  the  amount  he  wanted.     How  many  cents'  worth 
of  ice  cream  did  he  want  ? 

17.  Jennie  has  7  cents,  which  is  25%  of  her  sister's  money, 
and  50%  of  her  brother's  money.     How  many  cents  has  each 
of  them  ? 

18.  $  45.75  is  25%  of  how  many  dollars  ? 

19.  How  many  pounds  in  25%  of  a  ton  ?    In  10%  ?     20%  ? 

20.  Mr.  Wade  invested  $870,  and  gained  10%  on  it  in  a 
year.     How  much  had  he  at  the  end  of  the  year  ? 

21.  Mr.  Brooks  invested  $9000,  gained  10%  on  it  in  the 
first  year,  and  added  the  gain  to  his  capital.     He  gained  10% 


MISCELLANEOUS  EXERCISES  53 

on.  that  amount  in  the  second  year,  and  added  it  to  his  capital. 
During  the  third  year  he  increased  his  capital  by  10%.  Find 
how  much  he  had  at  the  end  of  each  year.  How  much  more 
than  his  original  investment  had  he  at  the  end  of  the  third 
year? 

22.  How  much  is  100%  of  2  watermelons  ?     Of  4  chairs  ? 

23.  A  chair  that  cost  $  3  was  sold  at  a  gain  of  100%.     For 
how  much  was  it  sold  ? 

24.  What  number  is  as  much  greater  than  10  as   10  is 
greater  than  8  ? 

25.  What  number  is  as  much  more  than  20  as  20  is  more 
than  17  ?     As  much  less  than  20  as  20  is  less  than  21  ? 

26.  What  is  the  average  of  10  and  16,  or  what  number  is  as 
much  greater  than  10  as  it  is  less  than  16  ? 

To  find  the  average  of  two  numbers,  divide  their  sum  by  2.  To  average 
three  numbers,  divide  their  sum  by  3.  To  average  four  numbers,  divide 
their  sum  by  4,  etc. 

27.  What  is  the  average  of  18  and  20  ?     Of  4  and  50  ?     Of 
9, 21,  and  24  ?     Of  8,  12,  and  25  ? 

28.  If   you    stand  98%   on   an   arithmetic   test,    95%    on 
a  spelling  test,  and  92%  on  a  geography  test,  what  is  your 
average  per  cent  ? 

29.  Joseph  worked  9  problems  on  Monday,  12  on  Tuesday, 
and  12  on  Wednesday.     How  many  problems  a  day  did  he 
average  ? 

30.  What  is  the  average  of  8.48,  10.24,  and  4.96  ? 

31.  Mr.   Harris   earned  $25.37  in  one  week,  $38.75  the 
next  week,  $  31.25  the  next  week,  and  $  40.50  the  next  week. 
How  much  were  his  average  earnings  during  the  four  weeks  ? 

32.  High-water  mark  at  a  certain  town  on  a  large  river  was 
48.3  ft.  one  year,  50.5  ft.  the  next,  and  47.6  ft.  the  next.     WJiat 
was  the  average  ? 


54  INTEGERS   AND   DECIMALS 

33.  What  was  the  average  height  of  a  river  for  four  suc- 
cessive days,  if  on  the  first  day  it  was  33.9  ft.  high,  on  the  next 
34.3  ft.,  on  the  next  34.9  ft.,  and  on  the  next  35.1  ft.  ? 

34.  Mr.  Howe  invested  $  36,000  in  business.     At  the  end  of 
8  years  his  capital  was  $  64,000.     What  was  his  average  gain 
per  year  ? 

NOTE  TO  TEACHER.     The  following  work  requires  a  Fahrenheit  ther- 
mometer and  an  explanation  of  its  use. 

35.  How  many  degrees  are  there  between  the  freezing  point 
and  the  boiling  point  on  the  Fahrenheit  thermometer  ? 

36.  The  temperature  at  noon  for  four  successive  days  in 
July  was  90°,  91°,  92°,  and  84°.     Find  the  average  temperature. 

37.  The  temperature  at  noon  for  five   successive   days   in 
January  was  21°,  19°,  15°,  18°,  and  27°.     Find   the   average 
temperature. 

38.  The  temperature  of  our  bodies  is  about  98°.     How  much 
above  freezing  point  is  that  ?     Below  the  boiling  point  ? 

39.  How  many  degrees  does  the  temperature  fall  when  it 
changes  from  57°  to  3°  below  the  freezing  point  ? 

*  40.   What  is  the  decrease  in  temperature  from  20°  to  1° 
above  0  ?     From  20°  to  1°  below  0  ?     From  15°  to  5°  below  0  ? 

41.  The  temperature  at  Minneapolis  one  winter  day  was  11°. 
Before  night  it  fell  20°.     What  was  the  temperature  then  ? 

42.  The  next  day  it  rose  15°.     What  was  the  temperature? 

43.  The  next  morning  it  was  4°  below  0.     How  much  had 
it  fallen  ? 

44.  Mrs.  A.  bought  19^  worth  of  groceries  and  offered  $  1.00 
in  payment.     The  clerk   gave  her   1^   and   said,   "  Twenty." 
Then  he  gave  her  a  nickel  and  said,  "  Twenty-five."     He  then 
gave  her  a  quarter  saying,  "  Fifty."     He  ended  by  giving  her  a 
half  doliar  and  saying,  "  One  dollar."     In  the  same  way  find 
how  that  amount  of  change  could  be  given  with  different  coins. 


MISCELLANEOUS   EXERCISES  55 

45.  CLASS  EXERCISE.     may  tell  a  story  of  a  purchase 

made  and  payment  offered.     Members  of  the  class  may  show 
different  ways  of  making  change. 

46.  Draw  two  horizontal  lines  and  two  vertical  lines. 

47.  Lines  which  lie  in  the  same  direction  are  called  Parallel 
Lines.     Find  parallel  lines  in  the  room.     In  your  book.     Name 
some  capital  letters  that  have  parallel  lines  when  printed. 

48.  Think  of  your  own  name  printed  in  capitals.     Can  you 
see  any  parallel  lines  in  it  ? 

49.  How  many  pairs  of  parallel  lines  has  a  rectangle  ?     Are 
there  any  parallel  lines  in  a  triangle  ? 

A      4.5     B  50.    A  four-sided  figure  that  has  only  2 

^  ^\    parallel  sides  is  called  a  Trapezoid.     AB 

c  and  CD  are   parallel.     How  long   is  the 

perimeter   of   the    trapezoid   ABCD,    the 

measurements  representing  inches  ? 

51.  The  sum  of  the  parallel  sides  of  Fig.  12  is  how  much 
more  than  the  sum  of  the  non-parallel  sides  ? 

52.  Draw  trapezoids  of  different  shapes. 

53.  Two  lines  meeting  at  a  point  form  an  Angle,  Z.     The 
point  where  the  lines  meet  is  called  the  Vertex  of  the  angle. 
Draw  an  angle  and  mark  its  vertex  A. 

A  54.  When  one  straight  line  meets  another 

straight  line  so  as  to  make  two  equal  angles, 
the  angles  are  called  Right  Angles.  What 
letter  is  at  the  vertex  of  each  angle  in 
Fig.  13? 

Right  55t   Place  two  pencils  so  as  to  show  two 


Right 
Angle 


Angle         right  angles. 


56.    Fold  a  strip  of  paper  so  that  the 
FIG.  13.  line  of  the  fold  makes  right  angles  with 

1  the  edge. 


56  INTEGERS  AND  DECIMALS 

57.  Cut  out  a  paper  circle  and  fold  it  into  fourths.     What 
kind  of  angles  are  made  by  the  folds  ? 

58.  Find  right  angles  made  by  lines  in  the  surfaces  of  the 
room  or  of  objects  in  it. 

59.    An  angle  less  than  a  right  angle  is  called 
an  Acute  Angle.     Draw  an  acute  angle. 

SUGGESTION    TO   TEACHER.      As  children   naturally 
FIG.  14.  Judge  of  the  size  of  an  angle  by  the  length  of  the  lines 

that  form  it,  pupils  should  draw  and  cut  out  a  right 
angle,  and  by  applying  it  to  given  angles,  find  out  whether  they  are 
acute,  right,  or  obtuse. 

60.  Draw  a  trapezoid  and  mark  the  acute  angles. 

61.  An  angle  greater  than  a  right  angle 
is  called  an  Obtuse  Angle.  Draw  an  obtuse 
angle. 


FIG.  15.  62.    Draw  a  trapezoid  and  mark  obtuse 

angles  and  acute  angles. 

63.  What  kind  of  angles  has  a  rectangle  ?    An  equilateral 
triangle  ? 

64.  Draw  a  trapezoid  that  has  two  right  angles.     Name  the 
other  two  angles. 

65.  In  the  printed  words  "ADMIRAL  DEWEY,"  how  many 
right  angles  are  there  ?     Acute  angles  ?     Obtuse  angles  ? 

66.  If  the  name  of   the  county  in  which  you   live  were 
printed  in  Gothic  type  like  the  words  "  ADMIRAL  DEWEY," 
how  many  right  angles  would  there  be  in  it?     How  many 
acute  angles?     How  many  obtuse  angles? 

SUGGESTION  TO  TEACHER.    Splints  or  toothpicks  are  useful  in  the 
following  exercises. 

67.  With  3  lines  make  2  right  angles;   2  obtuse  angles; 
2  acute  angles.     Show  the  vertices  of  the  angles. 


MISCELLANEOUS   EXERCISES 


57 


68.  With  2  lines  make  4  angles,  and  tell  of  what  kind  they 
are. 

69.  With  3  lines  make  12  angles,  and  tell  their  kinds.    Make 
10  angles.     9  angles. 

70.  With  4  lines  make  16  angles.     20  angles.     24  angles. 

71.  With  5  lines  make  4  angles.     5  angles.     20  angles. 


72.    CLASS  EXERCISE. 


may  tell  how  many  angles  he 


can  make  with  a  certain  number  of  lines,  and  the  class  may 
make  them. 

73.  A  triangle   that  has   a  right  angle  is  called  a  Right 
Triangle.     Draw  a  right  triangle. 

74.  Draw  an  isosceles  triangle.     The  angles  at  the  base  are 
equal.     What  kind  of  angles  are  they  ? 

75.  Draw  an  isosceles  triangle  on  paper.     Cut  it  out  and 
fold  it  so  that  the  equal  sides  coincide.     Cut  along  the  line  of 
the  fold,  and  you  have  two  equal  triangles.     What  kind  of  tri- 
angles are  they  ? 


76.  How  long  would  the  perimeter  of  one  of  these  right  tri- 
angles be  if  the  base  were  20  in.,  perpendic- 
ular 11  in.  longer  than  the  base,  and  hypot- 
enuse 8  in.  longer  than  the  perpendicular  ? 


<< 


Base 


FIG.  16. 


77.  Place  together  the  two  triangles 
you  have  made  so  that  they  form  a 
rectangle.  If  the  area  of  that  figure 
were  28  sq.  in.,  what  would  be  the  area 
of  each  right  triangle  ? 


78.  A  figure  drawn  upon  a  flat  surface  is  called  a  Plane 
Figure.  Can  you  draw  a  plane  figure  on  the  surface  of  a  ball  ? 
Of  a  slate  ?  Of  a  piece  of  gas  pipe  ? 


58 


INTEGERS  AND   DECIMALS 


FIG.  17. 


79.  A  plane  figure  bounded  by  five  straight 
lines  is  called  a  Pentagon.     When  (as  in 
Fig.  17)  the  lines  are  all  equal  and  make 
equal  angles,  the  figure  is  called  a  Regular 
Pentagon.     What    kind   of    angles    has   a 
regular  pentagon? 

80.  Find  the  length  of  the  perimeter  of 
the  pentagon  represented  by  Fig.  17. 

81.    How  long  is  one  side  of  a  regular  pentagon  whose  perim- 
eter is  9.15  in.  ? 

82.  Draw   a    pentagon    that    is    not 
regular. 

83.  The    pentagon    in    Fig.    18    is 
o   divided   into  triangles   by   equal   lines 

drawn  from  its  center  to  the  vertices 
of  its  angles.  What  kind  of  triangles 
are  thus  formed  ?  How  many  of  them  ? 
Each  triangle  is  what  part  of  the  penta- 
gon? What  %? 


FIG.  18. 


84.  What  %  of  the  pentagon  is  the  figure  ABCO  ?    AEDO  ? 
BCDEO?     CDEABO? 

85.  Figure  19  differs  from  Fig.  18  in 
having  the  lines  Og,  Oh,  etc.,  drawn 
from  the  center  of  the  pentagon  to 
the  middle  point  of  each  side.  They 
are  perpendicular  to  the  sides.  Each 
right  triangle  thus  formed  is  what  part 
of  the  pentagon  ?  What  %  ? 


E  i  D 

FIG.  19. 


86.  What  %  of  the  pentagon  is  AOB? 
EOi?     BOhC?    ABCh? 


87.    Give  the  outlines  of  a  figure  which   is  70%   of  the 
pentagon. 


MISCELLANEOUS  EXERCISES  59 

88.  A  butcher  bought  a  hog  weighing  375  Ib.  at  $  .03  a  Ib. 
How  much  did  it  cost  ? 

89.  He  sold  15  Ib.  of  it  at  2^  per  Ib.,  50  Ib.  at  5^  and  the 
rest  at  12^  per  Ib.     How  much  did  he  receive  for  it  ? 

90.  A  farmer  sold  15  doz.  eggs  at  18^  a  dozen,  receiving  for 
them  sugar  at  6^  a  pound.     How  many  pounds  of  sugar  did 
he  receive? 

91.  A  fruit  dealer  buys  29  doz.  oranges  for  $8.70.     How 
much  does  he  pay  for  each  orange  ? 

92.  If  he  sells  the  oranges  at  the  rate  of -5^  apiece,  how 
much  does  he  gain  on  each  orange  ?     How  much  on  all  the 
oranges  ? 

93.  What  number  multiplied  by  9  will  give  the  same  prod- 
uct as  12  multiplied  by  6  ? 

94.  Mr.  Hale  had  $  5728  and  paid  25%  of  it  for  a  farm. 
How  much  did  the  farm  cost  ?     He  sold  the  farm  for  $  1200. 
How  much  did  he  lose  ?     ^  ; 

IVA  A    ' 

^  95.  He  left  the  other  75p  of  his  money  in  the  bank  until  it 
had  gained  $472  interest.  How  much  money  had  he  then, 
including  the  money  he  received  from  his  farm  ? 

96.  A  grocer  bought  185  barrels  of  flour  at  $  3.75  a  barrel, 
and  sold  it  all  for  $  740.     How  much  did  he  gain  ? 

97.  A  miller  bought  35  bu.  of  wheat  for  $  22.75,  and  sold  it 
at  $  .61  a  bushel.    Row  much  did  he  lose  ? 

98.  A  farmer  had  an  orchard  of  276  trees.     One  year  they 
averaged  13  bu.  of  apples  to  each  tree.     What  was  the  value 
of  that  season's  crop  at  $  .75  a  bushel  ? 

99.  The  next  year  the  trees  averaged  9  bu.  per  tree,  and 
the  apples  brought  $  .80  a  bushel.     What  was  the  value  of  the 
crop  that  year  ? 


60  INTEGERS  AND  DECIMALS 

100.  A  merchant's  profits  in  January,  1899,  were  $  1428.75. 
In  January,  1900,  his  profits  were  20  %  less.     What  were  his 
profits  in  January,  1900  ? 

101.  Mr.  Strong  had  $975.85  in  a  bank;  he  drew  a  check 
on  the  bank  for  $  625.47.     How  much  money  had  he  remain- 
ing in  the  bank  ? 

SUGGESTION  TO  TEACHER.     Show  bank  checks.     Explain  their  use  and 
let  pupils  copy  and  fill  them  out  for  use  in  imagined  transactions. 

102.  If  you  had  $  65.87  in  a  bank,  and  should  draw  a  check 
for  $  38.45,  how  much  of  your  money  would  be  left  in  the  bank  ? 

103.  Mr.  Gale  had  $1225  in  a  bank.     He   drew   $12.25 
every  Saturday  night  for  10  weeks.     How  much  had  he  left  in 
the  bank  ? 

104.  Aline  deposited  $  11.75  in  a  savings  bank  in  February. 
She  drew  out  $  3.25  in  March  and  $  2.95  in  April.     She  de- 
posited $  14.45  in  May.    How  much  had  she  then  in  the  bank  ? 

105.  Mr.  Davis  bought  a  stove  worth  $  18.75.     The  dealer 
allowed  him  2^  a  pound  for  an  old  stove,  weighing  195  Ib.    He 
gave  a  check  on  the  bank  for  the  balance.     What  was  the 
amount  of  the  check  ? 

106.  There  were   276   houses   on   the   street.     A  postman 
delivered  3  letters  at  28  of  the  houses,  2  letters  at  41  of  the 
houses,  and  1  letter  at  105  houses.     At  how  many  houses  were 
no  letters  delivered  ? 

107.  There  were  559  books  in  a  school  library,  which  was 
an  average  of  13  to  each  pupil.     How  many  pupils  were  there 
in  the  school  ? 

108.  In  February  of   a  common  year,   Mr.   Fisk's  family 
burned  a  ton  of  coal  in  14  days.     At  $  8.50  per  ton,  what  was 
the  cost  of  the  coal  for  that  month  ? 

109.  A  lot  is  in  the  form  of  a  trapezoid.     One  of  the  parallel 
sides  is  16.8  rd.  long,  and  the  other  is  twice  as  long.     Of  the 


MISCELLANEOUS  EXERCISES  61 

sides  that  are  not  parallel,  one  is  19.7  rd.  long,  the  other  is 

15.4  rd.  long.     How  long  a  fence  is  required  for  the  whole  lot  ? 

110.  Mr.  Lee  started  to  Denver  with  $  300.     He  paid  $  47 
for  railroad  fare,  his  hotel  bill  was  $  4  a  day  for  a  week,  other 
expenses  $  7.50,  and  his  return  ticket  was  $  47.     How  much 
did  he  spend  ?     How  much  had  he  left  ? 

111.  Passengers  were  first  carried  on  railroads  in  the  United 
States  in  the  year  MDCCCXXVIIL     How  many  years  have 
we  had  railroads  ? 

112.  Square:  13.     1.5.     1.7. 

113.  Ella  had  a  flower  bed  a  yard  square.     She  divided  it 
into  square  feet  and  placed  a  rosebush  in  the  middle  of  each 
square  foot.     How  many  rosebushes  had  she  ?     Represent. 

114.  Draw  a  square  decimeter   and   show  into  how  many 
square  centimeters  it  can  be  divided.     Into  how  many  square 
inches  can  a  square  foot  be  divided  ? 

115.  How  many  square  inches  in  5  squares  whose  sides  are 
each  11  in.  long  ? 

116.  How  long  is  the  perimeter  of  a  square,  a  side  of  which 
is  3.1  in.  long  ?     What  is  its  area  ? 

117.  Draw  a  right  triangle.      If  its  base  were  7.5  in.,  its 
perpendicular  10  in.,  and  its  whole  perimeter  30  in.,  how  long 
would  the  hypotenuse  be  ?     What  would  be  its  area  ? 

118.  Find  the  length  of  the  perpendicular  of  a  right  triangle 
whose  perimeter  is  90  in.,  its  base  22.5  in.,  and  hypotenuse 

37.5  in.     Find  the  area  of  the  triangle. 


CHAPTER   II 
PROPERTIES   OF  NUMBERS 

1.  When   an   integer  can  be   divided   by  another  number 
without  a  remainder,  it  is  said  to  be  divisible  by  that  number. 
Is  9  divisible  by  5  ?     Give  a  reason  for  your  answer. 

2.  Choose  an  even  number  and  illustrate  this  statement : 
An  even  number  is  an  integer  that  is  divisible  by  2. 

3.  Choose  an   odd   number  and   illustrate  this  statement: 
An  odd  number  is  an  integer  that  is  not  divisible  by  2. 

4.  Name  the  first  even  number  after  10.     How  many  twos 
does  it  equal  ? 

5.  Square  the  third  odd  number.      Square  the  sixth  even 
number.     Multiply  the  seventh  even  number  by  the  fifth  odd 
number.     Find  the  difference  between  the  sixth  odd  number 
and  the  eighth  even  number. 

6.  Find  |  of  the  fifth  even  number.    Find  7%  of  the  fourth 
odd  number. 

MULTIPLES  AND   FACTORS 

7.  A   Multiple   of   a   number   is   the   product   obtained   by 
multiplying  it  by  an  integer.     Thus  5  is  the  first  multiple 
of  5,  10  is  the  second  multiple  of  5.     Give  quickly  the  first 
twelve  multiples  of 

3        4        5        6        7        8        9        10        11        12 

8.  Figure  1  represents  two  lots  of  land  owned  by  Mr.  Smith 
and  Mr.  Brown.     The  fence  between  the  lots  is  150  ft.  long 


MULTIPLES   AND   FACTORS 


63 


and  cost  7^  a  foot.     How  much  of  this  expense  should  each 
man  pay  ?     Give  reasons  for  your  answer. 

. , _        9.    Mr.  Smith  and    Mr.  Brown  decided 

to  take  away  the  fence  and  leave  a  strip 
10  ft.  wide  on  each  side  of  the  line  where 
it  had  stood.  This  strip  is  used  as  a  com- 
mon playground  by  the  children  of  both 
families.  How  many  square  feet  in  their 
common  playground  ? 

10.    A  fine  park  in  Boston  is  called  Bos- 
ton Common.    What  does  the  word  "  com- 
mon "  mean  in  this  case  ?     "  20  is  a  com- 
What  does  that  statement  mean  ? 


FIG.  1. 


mon  multiple  of  10  and  5." 

11.  A  number  which  is  a  multiple  of  two  or  more  numbers 
is  called  their  Common  Multiple.     Give  several  numbers  that 
are  common  multiples  of  2  and  3.     Of  3  and  7.     Of  4  and  5. 

12.  Of  what  two  numbers  besides  itself  and  1  is  15  a  mul- 
tiple?    10?    35?     21?     22?     33? 

13.  Write  all  the  numbers  of  which  6  is  a  common  multiple. 
8.     14.     16.     12.     24.     40.     36. 

14.  Give  two  numbers  which  multiplied  together  make  the 
product  18.     20.     27. 

15.  The  numbers  that  make  a  product  are  called  the  Factors 
or  Divisors  of  that  product.     12  has  3  pairs  of  factors,  1  x  12, 
2x6,  and  3x4.     Give  all  the  pairs  of  factors  of  24,  except 
the  pair  of  which  1  is  the  least  factor. 

SUGGESTION  TO  TEACHER.  Let  pupils  find  factors  of  a  number  by 
using  as  a  trial  divisor  each  number  in  succession,  beginning  with  2. 
Lead  them  to  see  that  as  soon  as  the  quotient  which  they  obtain  is  less 
than  the  divisor  they  use,  it  is  unnecessary  to  try"  any  more  numbers,  as 
they  will  merely  get  the  same  pairs  of  factors  stated  in*  reverse  order. 

16.  Give  all  the  pairs  of  factors  of  45.     28.     36.     60.     72. 

17.  Give  all  the  numbers  of  which  30  is  a  multiple.     66. 


64  PROPERTIES   OF  NUMBERS 

18.  CLASS  EXERCISE.     may  give  a  number  which  is  a 

multiple  of  some  other  numbers,  and  the  class  may  find  all 
its  factors. 

COMPOSITE  NUMBERS 

19.  A  number  that  is  the  product  of  two  or  more  integers 
is  called  a  Composite  Number.     Give  three,  composite  numbers 
and  their  factors. 

20.  What  number  is  composed  of  the  factors  2  and  11  ?    3 
and  11  ?     2,  3,  and  11  ?     7  and  7  ?     2,  3,  and  7  ?     2,  2,  and  3  ? 

21 .  What  factors  compose  77  ?     40  ?     18  ?     42  ? 

22.  Name  an  even  composite  number,  and  give  factors  that 
compose  it. 

23.  Name  an  odd  composite  number,  and  give  factors  that 
compose  it. 

24.  Name  a  composite  number  that  is  a  multiple  of  5,  and 
give  its  other  divisors. 

25.  Make  and  keep  a  list  of  all  the  composite  numbers  less 
than  41. 

26.  "Write  a  composite  number  whose  tens'  digit  is  2,  and 
give  its  factors. 

27 .  Write  the  following  numbers  and  their  factors : 
a  A  composite  number  whose  units'  figure  is  5. 

I)  The  first  composite  number  after  26. 

• 

c  A  composite  number  between  30  and  40  that  is  not  a  mul- 
tiple of  5. 

d  A  composite  number  between  30  and  40  that  is  not  a  mul- 
tiple of  2  nor  of  5. 

e  A  composite  number  between  20  and  30  that  is  not  a  mul- 
tiple of  2,  5,  nor  7. 

28.  Divide  .00168  by  the  3d  composite  number. 


PRIME  NUMBERS  65 

29.  Find  5%  of  the  9th  composite  number. 

30.  Multiply  the  8th  composite  number  by  .009. 

PRIME  NUMBERS 

31.  A  number  that  has  no  integral  factors  except  itself  and 

1  is  called  a  Prime  Number.    Think  of  each  of  the  numbers  from 

2  to  10  and  tell  which  of  them  are  prime. 

2  is  the  first  prime  number,  as  1  is  considered  neither  prime  nor 
composite. 

32.  No  prime  number  of  more  than  one  place  ends  in  2,  4,  6, 
8,  0,  or  5.     Can  you  tell  why  ? 

33.  Make  a  list  of  all  the  prime  numbers  less  than  50  in 
their  order. 

SUGGESTIONS  TO  TEACHER.  Show  pupils  how  to  find  prime  numbers 
less  than  50  by  examining  each  number  to  see  if  it  can  be  exactly  divided 
by  2,  3,  5,  or  7.  Develop  the  fact  that  there  is  no  need  of  dividing  even 
numbers  because  they  are  all  divisible  by  2  ;  nor  numbers  that  end  in  5, 
because  they  are  divisible  by  5.  Lead  pupils  to  see  that  if  a  number  will 
not  contain  2,  it  will  not  contain  4,  6,  8,  or  any  number  of  twos ;  that  if 
it  will  not  contain  3,  it  will  not  contain  9,  15,  21,  or  any  number  of  threes, 
and  so  on. 

'  34.    Group  the  prime  numbers  less  than  50  as  they  are  found 
in  each  ten  numbers,  as : 


1st  ten 


2dten 


11 
13 
17 


Keep  the  list. 


19 

35.  Find  the  sum  of  all  the  prime  numbers  that  are  ex- 
pressed by  one  digit. 

36.  Divide  13.5  by  the  2d  prime  number. 

37.  Divide  the  3d  prime  number  by  .8. 

38.  Find  the  difference  between  26.4  and  the  llth  prime 
number. 

HORN.    GRAM.    SCH.    AR.  —  5 


66  PROPERTIES  OF  NUMBERS 

39.  Multiply  the  8th  prime  number  by  .0004. 

40.  Find  6%  of  the  9th  prime  number. 

41.  Write  the  first  30  numbers  in  two  lists,  one  of  prime 
numbers,   the  other  of  composite   numbers.      Leave  out  the 
number  1. 

42.  Find  the  sum  of  all  the  composite  numbers  less  than  15. 

43.  Find  the  sum  of  all  the  primes  less  than  20. 

44.  What  prime  number  is  nearest  to  20  ? 

45.  What  two  prime  numbers  are  near  to  12? 

46.  15  is  half  way  between  two  prime  numbers.     What  are 
they  ? 

There  are  three  pairs  of  these  numbers. 

47.  What  prime  number  is  nearest  to  the  2d  multiple  of  5  ? 
To  the  8th  multiple  of  5  ? 

SUGGESTION  FOR  CLASS  EXERCISE.  Let  children  select  prime  numbers 
and  give  a  clew  to  them,  and  let  the  class  identify  them. 

48.  Find  the  difference  between  the  prime  number  nearest 
to  20  and  the  prime  number  nearest  to  8. 

49.  CLASS  EXERCISE.     may  name  a  number  larger  than 

50  which  he  thinks  is  prime,  and  the  class  may  see  if  he  is  right. 

NOTE  TO  TEACHER.  The  following  method  of  finding  prime  numbers 
less  than  100  is  very  useful : 

Write  the  first  hundred  numbers  as  on  p.  67,  omitting  1  because  it  is  con- 
sidered neither  prime  nor  composite.  Counting  from  2,  the  first  prime 
number,  strike  out  as  composite  every  second  number  because  it  is  a 
multiple  of  2  ;  counting  from  3,  strike  out  as  composite  every  third 
number.  Lead  the  pupils  to  discover  why  it  is  unnecessary  to  strike  out 
the  multiples  of  11  or  of  any  larger  primes  in  finding  the  prime  numbers 
less  than  100. 

This  device,  which  is  an  adaptation  of  the  well-known  "sieve  of 
Eratosthenes,"  may  be  used  to  any  limit  to  separate  prime  from  compos- 
ite numbers  by  writing  additional  columns  of  numbers  and  striking  out  all 
the  multiples  except  the  first  multiple  of  those  primes  whose  squares  are 
included  within  the  limit. 


t      '  n  f\  pr 

PRIME  NUMBERS  67 

.- 

11         2;         31         41         $1         61         71         2J  91 

2T9  99  319  AW  39  ft0'  79'  S9  Q9 

A^  ^  P^  rr  Pr  rr  l*r  Pr  PP 

3         13         23         33         43         53         03         73         83          ?3 

7         17         27         37       .47         57         67         77         27  97 

£        1%       ?$        32        £2        ^2        02        72        22         R2 
?         19         29         3?         £9         59         0?         79         89  99 

50.  A  Greek  mathematician  named  Eratosthenes,  who  was 
born  275  B.C.,  devised  this   plan  of   finding   prime   numbers. 
Instead  of   marking   out   the   composites,   he   cut   them   out. 
Can  you  see  why  the  table  of  primes  that  was  left  was  called 
"  Eratosthenes'  sieve  "  ? 

51.  How  many   and   what  prime  numbers   in  the  2d  ten 
numbers  ?     In  the  10th  ten  ?     In  the  5th  ten  ? 

52.  Give  the  primes  less  than  100  whose  units'  digit  is  1. 
3.     7.     9. 

53.  Find  the  sum  of  all  the  primes  in  the  3d  ten.     In  the 
6th  ten.     In  the  4th  ten.     In  the  7th  ten. 

54.  Name  all  the  prime  numbers  less  than  100  whose  tens' 
digit  is  2.     4.     1.     3.     5.     7.     9.     6.     8. 

NOTE  TO  TEACHER.  This  adaptation  of  "Eratosthenes'  sieve"  may  be 
made  helpful  in  studying  composite  numbers,  divisors,  and  multiples. 

Let  the  columns  of  numbers  be  written  on  the  board  in  large  figures. 
Instead  of  striking  out  multiples,  draw  a  circle  or  square  around  each  of 
them,  using  crayon  of  the  same  color  to  inclose  the  multiples  of  a  partic- 
ular number.  If,  for  instance,  the  multiples  of  2  are  inclosed  in  blue, 
those  of  3  in  red,  those  of  5  in  green,  those  of  7  in  yellow,  30  shows  itself 
at  once  by  its  motley  framing  as  a  multiple  of  2,  3,  and  5  ;  42  by  its 
slightly  different  framing  as  a  multiple  of  2,  3,  and  7  j  66  as  a  multiple  of 
2,  3,  and  11. 


68  PROPERTIES   OF  NUMBERS 

The  following  questions  are  based  upon  a  diagram  of  that  kind  and 
refer  to  numbers  less  than  101. 

55.  Point  out  the  multiples  of  3  whose  units'  digit  is  2.     3. 
5.     6.     7.     8.     9.     4. 

56.  Point  out  the  multiples  of  7  whose  units'  digits  is  1.     2. 
4.     9. 

57.  What  figure  ends  the  expression  of  all  the  multiples  of 
5  that  are  odd  numbers  ?     Even  numbers  ? 

58.  Point  out  all  the  numbers  that  are  multiples  of  2  and 
also  of  5,  beginning  with  the  least. 

59.  Show  the  common  multiples  of  2  and  3,  beginning  with 
the  least  common  multiple. 

60.  Show  the  common  multiples  of  2  and  7,  beginning  with 
the  least  common  multiple. 

61.  Beginning  with  the  least  common  multiple,  show  all  the 
common  multiples  of  3  and  7.     Of  3  and  5.     Of  5  and  7.     Of 
2,  3,  and  5.    Of  2,  3,  and  7.    Of  2,  5,  and  7. 

62.  CLASS  EXERCISE.     The  teacher  or  a  pupil  pointing  to  a 
number  in  the  diagram,  members  of  the  class  tell  of  what 
numbers  it  is  a  common  multiple,  and  whether  or  not  it  is 
their  least  common  multiple. 


PRIME   FACTORS 

63.  Those  factors  of  a  number  that  are   prime,  are  called 
Prime  Factors.    What  are  the  prime  factors  of  4  ?    Of  6?    Of  8? 

64.  Find  the  prime  factors  of  42. 

2 1 42          To  find  the  prime  factors  of  a  number  divide  it  by  the  smallest 
3  "21      prime  number  of  which  it  is  a  multiple.     Then  divide  the  quo- 
4r      tient  by  the  smallest  prime  number  of  which  it  is  a  multiple. 
Continue  dividing  until  the  quotient  is  prime.     In  this  case  42 
divided  by  2  gives  a  quotient  of  21,  21  divided  by  3  gives  the  prime  quo- 
tient 7,     Hence  the  prime  factors  of  42  are  2,  3,  and  7. 


LEAST  COMMON  MULTIPLE  69 

65.  Find  the  prime  factors  of: 

10  15  20  24  27  32  35  39  44  48 
12  16  21  25  28  33  36  40  45  49 
14  18  22  26  30  34  38  42  46  50 

66.  Find  the  prime  factors  of  all  the  even  numbers  greater 
than  49  and  less  than  59.     Of  all  the  composite  odd  numbers 
between  those  limits. 

67.  Find  the  prime  factors  of  all  the  even  numbers  between 
59  and  69.     Of  all  the  composite  odd  numbers. 

68.  Find  the  prime  factors  of  all  the  multiples  of  5  between 
69  and  91.     Of  all  the  multiples  of  3  between  those  limits.     Of 
all  the  composite  numbers  between  90  and  101. 

69.  Resolve  into  prime  factors : 

a  336  c  1225         e  639        g  3105        t  1470        fc  1296 

b  3456         d  2214        /  560        h  888          j  810  I  1488 

70.  CLASS  EXERCISES.     may  give  to  the  class  a  number 

that  is  the  product  of  several  small  prime  numbers,  and  the 
class  may  find  its  prime  factors. 

71.  Separate  the  first  100  numbers  into  two  lists,  one  of 
prime  numbers,  the  other  of  composite  numbers.     Write  oppo- 
site each  composite  number  the  prime  factors  of  which  it  is 
composed. 

SUGGESTION  TO  TEACHER.     Class  drill  upon  these  lists  should  be  given 
frequently  until  their  contents  are  learned. 

LEAST  COMMON  MULTIPLE 

72.  Forty  contains  how  many  more  fives  than  eights  ? 

73.  CLASS  EXERCISE.     '-  may  name  a  number  which  is 

a  common  multiple  of  two  or  more  numbers.     The  class  may 
give  the   numbers   and  tell  how  many  times  their  common 
multiple  contains  each  of  them. 


70  PROPERTIES  OF  NUMBERS 

74.  The   Least   Common    Multiple  of    two   or    more   prime 
numbers  is  their  product ;  the  next  common  multiple  is  twice 
their  product ;  the  next  is  three  times  their  product.     What  is 
the  next  ?     Write  the  first  six  common  multiples  of  2  and  5, 
and  underscore  the  least. 

75.  Write  the  first  four  common  multiples  of  3  and  5.    Give 
the  least  common  multiple  and  find  how  many  times  3  and  5 
are  each  contained  in  it. 

76.  Write   the   first   three  common  multiples  of  2  and  7. 
Find  how  many  times  2  and  7  are  each  contained  in  their  least 
common  multiple. 

77.  Find  the  least  common  multiple  of  2,  3,  and  5.     How 
many  threes  does  it  contain  ?     How  many  twos  ?     How  many 
fives? 

78.  Find  the  least  common  multiple  of  2,  3,  and  7.     How 
many  sevens  in  it  ?     Twos  ?     Threes  ? 

79.  The  abbreviation  for  least  common  multiple  is  1  .c.  m. 
Find  the   1.  c.  m.   of  2,  5,  and  7.      How   many   fives   in   it  ? 
Sevens  ?     Twos  ? 

80.  Find  the  1.  c.  m.  of  5  and  9.     Use  the  following  method 
of  finding  the  1.  c.  m.  mentally :     Think  of   the   multiples  of 
the  larger  number  in  order   until  one  is  found  which  is   a 
multiple  of  the  smaller.     For  instance,  in  finding  the  1.  c.  m. 
of  5  and  9,  think  of  the  multiples  9,  18,  27,  36,  until  the  first 
that  contains  5  is  reached. 

81.  Find  the  1.  c.  m.  of : 

a  10  and  3  10  and  5  10  and  8  10  and  12  10  and  15 
b  6  and  9  4  and  9  7  and  9  11  and  9  8  and  9 
c  8  and  3  8  and  5  8  and  6  8  and  10  8  and  11 

82.  Find  the  1.  c.  m.   of  6,  5,  and  3.     Can  a  number  be  a 
multiple  of  6,  without  being  also  a  multiple  of  3  ? 


LEAST   COMMON  MULTIPLE  71 

83.  Find  the  1.  c.  m.  of : 

a  6,  2,  and  3  10,  5,  and  2  20,  10,  and  5  7,  5,  and  6 
b  10,  5,  and  6  8,  4,  and  12  5,  10,  and  15  8,  4,  and  3 
c  2,  3,  4,  and  6  3,  4,  8,  and  6  4,  8,  and  7  3,  6,  and  8 

84.  12  is  the  1.  c.  m.  of  2,  3,  6,  and  4.     Make  similar  state- 
ments about  the  numbers  18,  20,  24,  25,  30,  36,  35,  and  48. 

85.  Which  of  the  first  12  multiples  of  3  are  common  mul- 
tiples of  12  and  3  ? 

86.  Can  you  find  the  greatest  common  multiple  of  3  and  4? 
Of  any  two  other  numbers  ?     Explain. 

SUGGESTION  TO  TEACHER.     Some  of  the  pupils  inay  discover  that  the 
search  for  the  greatest  common  multiple  leads  into  infinity. 

87.  Draw  a  line  18  in.  long,  and  show  how  many  times  a 
3-in.  line  can  be  laid  off  upon  it.     How  many  times  can  a  6-in. 
line  be  laid  off  upon  it  ? 

88.  How  long  is  the  shortest  line  that  can  be  laid  off  into 
2-in.    lines   or   7-in.  lines?     3-in.   lines   or   7-in.  lines?     7-in. 
lines  or  5-in.  lines  ?     5-in.  lines  or  11-in.  lines  ? 

89.  How  large  is  the  smallest  number  that  can  be  divided 
into  groups  of  2  and  of  7  ?     Into  groups  of  3  and  of  7  ?     Into 
groups  of  7  and  of  5  ?     Into  groups  of  5  and  of  11  ? 

90.  CLASS  EXERCISE. may  name  a  composite   num- 
ber, and  others  may  show  the  different  groups  into  which  it 
may  be  separated. 

91.  A  company  of  children  were   playing   games.     At  first 
they  played  games  which  required  them  to  be  divided  into 
groups  of  three.     Afterwards  they  played  in  groups  of  four. 
Every  child  played  all  the  time.     What  is  the  least  number  of 
children  there  could  have  been  in  the  company  ? 

SUGGESTION  TO  TEACHER.     Illustrate,  by  grouping  children,  for  the 
benefit  of  those  pupils  who  cannot  imagine  clearly. 


72  PROPERTIES   OF   NUMBERS 

« 

92.  How  many  roses  must  a  girl  have,  to  be  able  to  divide 
them  into  bunches  of  3  roses  or  bunches  of  5  roses  ?     How 
many  bunches  may  she  have  of  3  roses  ?     Of  5  roses  ? 

93.  A  teacher  has  just  enough  pupils  to  divide  into  groups 
of  7  pupils  or  groups  of  4  pupils.     How  many  pupils  has  she  ? 
How  many  groups  of  each  kind  can  she  have  ? 

94.  I  have  just  enough  books  to  be  arranged»on  a  number 
of  shelves,  12  books  on  a  shelf,  or  by  using  more  shelves,  9 
books  on  each  shelf.     How  many  books  have  I  ?     How  many 
shelves  would  be  needed  under  the  first  arrangement  ?     Under 
the  second  ?    t 

95.  What    is   the   least    number   of    gallons  that  can   be 
exactly  measured  by  either  of  two  casks,  one  holding  6  gal., 
the  other  8  gal.?     How  many  times  can  the  smaller  cask  be 
filled  by  them  ?     The  larger  cask  ? 

96.  What   is  the  smallest  sum  of    money  that  can  be  en- 
tirely spent  in  buying  books  at  15  ^  apiece,  or  in  buying  books 
at  9^  apiece?     How  many  of  each  kind  of  books  could  be 
bought? 

97.  Ho.w  long  is  the  shortest  piece  of  ribbon  that  can -be 
cut  without  remainder  into  lengths  of  2  yd.,  3  yd.,  or  5  yd. 
each  ?     How  many  lengths  of  each  kind  could  be  made  ? 

98.  What  is  the  least  number  of  bananas  that  a  mother 
can    exactly   ivide   between   her   2   sons,   or   among    her    4 
daughters,  or  among  all  her  children  ?     How  many  bananas 
would  each  child  receive  in  each  case? 

NOTE  TO  TEACHER.  Written  methods  of  finding  the  1.  c.  in.  and 
g.  c.  d.  are  useful,  because  convenient,  but  the  reasons  for  the  processes 
are  beyond  the  comprehension  of  ordinary  children  in  the  grade  for 
which  this  work  is  designed.  Hence  the  methods  should  be  presented 
as  convenient  rules  that  have  been  discovered  by  mathematicians.  The 
reasons  for  these  rules  should  be  learned  later. 


'   9 


LEAST  COMMON  MULTIPLE  73 

99.    By  the  following  rule  find  the  1.  c.  m.  of  8  and  10 4 
To  find  the  least  common  multiple  of  several  numbers  — 
Arrange  the   numbers  in  a  horizontal  line,   leaving  out  all 
numbers  that  are  factors  of  any  of  the  other  numbers.     Find  the 
smallest  prime  number  that  will  exactly  divide  anyjwo^ofjhejn^ 
and  divide  by  it  all  the  numbers  of  which  it  is  a  factor,  placing 
the  quotients  and  undivided  numbers  below.     Continue  this  pro- 
cess until  no  prime  number  will  divide  two  o£  the  numbers  in  the 
last  horizontal  line.     Find  the  product  of  the  divisors,  quotients, 
ojid  undivided  numbers. 

100.   Find  the  I.e. m.  of  12,  16,  and  18. 


SOLUTION.     2 


12,  16,  18  2  x  2  x  3  x  4  x  3  =  144  1.  c.  m. 


6,    8,    9 


3,     4,     9 


1,     4,     3 

'  101.    Find  by  the  written  method  the  1.  c.  m.  of: 
a  25,  .60,  72,  35  c   63,  12,  84,  72  e  54,  81,  14,  63 

b   24,  16,  15,  20  d  16,  12,  24  /  15  9,  6,  5 

102.  Find  1.  c.  m.  of  5,  6,  18,  15. 

3)?,  0,  18,  15 

6,     5  3  x  6  x  5  =  90.  Ans. 

Why  may  the  5  and  6  be  crossed  out  and  not  considered  in  finding  the 
1.  c.  m.  of  5,  6,  18  and  15  ? 

103.  Find  1.  c.  m.  of: 

a  1,  2,  3,  4,  5,  6,  7,  8,  9  d  4,  8,  12,  24,  48,  84 

b    8,  12,  16,  24,  36,  48  e    5,  10,  15,  20,  30,  40 

c    4,3,6,7,8,16,9,  /    7,28,35,14,70 

104.  CLASS  EXERCISE.         —  may  name  four  composite  num- 
bers, and  the  class  may  find  their  1.  c.  m. 

105.  CLASS  EXERCISE.     may  name  three  prime  num- 
bers, and  the  class  may  find  their  1.  c.  m. 

106.  Numbers  that  are  multiples  of  any  given  number  are 
said  to  be  divisible  by  that  number.     Is  7  divisible  by  3  ? 
Name  several  numbers  that  are  divisible  by  10. 


74  PROPERTIES   OF   NUMBERS 

DIVISIBILITY  OF  NUMBERS 

107.  Illustrate  the  following  principle: 

PRINCIPLE  1.     A  number  that  ends  in  2,  4,  6,  8,  or  0  is  divisi- 
ble by  2. 

108.  Tell  without  dividing  which  of  the  following  numbers 
are  not  divisible  by  2,  and  what  the  remainder  is  in  each  case : 
8906.     2127.     2139.     2111.     2145.     1898. 

109.  Name  in  order  the  first  fourteen  multiples  of  5. 

110.  When  a  multiple  of  5  is  expressed  in  figures,  what 
figures  may  represent  the  units'  digit? 

PRINCIPLE  2.     A  number  that  ends  in  5  or  0  is  divisible  by  5. 

111.  Without  dividing,  select  from  the  following  the  num- 
bers that  are  not  divisible  by  5,  and  tell  what  the  remainder  is 
in  each  case:  75.     120.     81.     22500.     393.     920. 

112.  When  a  number  is  divisible  by  2  and  by  5,  with  what 
figure  must  its  expression  end  ?     By  what  other  number  is  it 
divisible  ? 

113.  Among  all  the  prime  numbers  less  than  100,  can  you 
find  any  the  sum  of  whose  digits  is  9  ? 

114.  Write  a  number  of  two  places  the  sum  of  whose  digits 
is  9.     Find  how  many  times  9  is  contained  in  that  number. 

115.  Write  a  number  of  three  places  the  sum  of  whose  digits 
is  9,  and  find  how  many  times  9  is  contained  in  that  number. 

116.  Write  a  number  of  four  places  the  sum  of  whose  digits 
is  9,  and  find  how  many  times  that  number  contains  9. 

117.  Write  a  number  the  sum  of  whose  digits  is  18,  and  see 
whether  it  contains  9  exactly. 

118.  Write  numbers  the  sum  of  whose  digits  is  9  or  some 
multiple  of  9,  and  divide  those  numbers  by  9  until  you  see  the 
truth  of  the  following  principle  : 


DIVISIBILITY  OF  NUMBERS  75 

PRINCIPLE  3.     Any  number  is  divisible  by  9  if  the  sum  of  its 
digits  is  a  multiple  of  9. 

119.  Tell  without   dividing  which  of  the  following  num- 
bers are  not  divisible  by  9,  and  give  the  remainder  in  each 
case:    2025.    105.    117.    112.    2114.    189.    207.    1026.    4154. 

120.  Write  several  large  numbers  of  which  9  is  a  divisor. 

121.  Write  ten  multiples  of  3  no  one  of  which  is  less  than 
36.     Find  the  sum  of  the  digits  of  each  of  them,  and  see  if 
that  sum  is  a  multiple  of  3. 

PRINCIPLE  4.     A  number  is  divisible  by  3  if  the  sum  of  its 
digits  is  a  multiple  of  3. 

122.  Tell  at  sight  which  of  the  following  numbers  are  not 
divisible  by  3,  and  give  the  remainder  in  each  case :  213.    411. 
6951.     343.     1125. 

123.  Can  you  find  a  multiple  of  9  that  is  not  a  multiple  of  3  ? 
Name  a  multiple  of  3  that  is  not  a  multiple  of  9. 

124.  Would  it  be  possible  for  a  number  to  be  a  multiple  of 
10  and  not  a  multiple  of  2  and  5  ?    Explain. 

125.  Choose  numbers  ending  in  0,  and  show  what  factors 
they  have  besides  2,  5,  and  10. 

126.  Can  a  number  be  divisible  by  6  without  being  divisible 
by  3  and  by  2  ? 

127.  Write  an  even  number  the  sum  of  whose  digits  is 
divisible  by  3.     As  the  number  is  divisible  by  3  and  by  2,  it  is 
divisible  by  6.     Write  three  other  numbers  divisible  by  6. 

128.  Write  three  numbers  each  of  which  is  divisible  by  3 
and  by  5.     Find  how  many  times  each  of  them  contains  15. 

129.  Write  three  numbers  ending  in  0  the  sum  of  whose 
digits  is  divisible  by  3.     Find  how  many  times  each  of  them 
contains  30. 


76  PROPERTIES    OF  NUMBERS 

130.  Write  three  numbers  divisible  by  2  and  by  9.     Divide 
each  of  them  by  18. 

131.  Write  three  numbers  divisible  by  5  and  by  9.     What 
number  between  40  and  50  is  a  factor  of  each  of  them  ?     How 
can  you  tell  ? 

132.  Count  by  4's  to  100. 

133.  Add  some  multiples  of  4  to  100,  and  see  if  the  num- 
bers thus  obtained  are  divisible  by  4. 

134.  Add  to  100  some  numbers  that  are  not  multiples  of  4, 
and  see  if  the  resulting  numbers  are  divisible  by  4.     Explain. 

PRINCIPLE  5.  A  number  is  divisible  by  4  if  the  number 
expressed  by  its  two  right-hand  figures  is  divisible  by  4. 

135.  Tell  without  dividing  the  whole  number  which  of  the 
following  numbers  are  not  divisible  by  4,  and  give  the  remain- 
der in  each  case:  127.    244.   365.    782.    728.    496.    338.    2672. 

SUGGESTION  TO  TEACHER.  Lead  pupils  to  see  that  as  100  is  a  multiple 
of  4,  any  number  of  hundreds  is  a  multiple  of  4,  and  if  there  is  added  to 
any  number  of  hundreds  a  number  which  is  an  aggregation  of  fours,  the 
result  will  be  a  still  greater  aggregation  of  fours. 

136.  Write   a  number   of  four   places.      Let   the   number 
expressed  by  the  two  right-hand  digits  be  a  multiple  of  4. 
Let  the  sum  of  the  digits  of  the  whole  number  be  a  multiple 
of  3.     Find  how  many  times  12  is  contained  in  the  number. 

137.  Write  three  numbers  divisible  by  4  and  by  5.     What 
multiple  of  10  besides  10  is  a  factor  of  each  of  them  ?    Explain. 

138.  Write  three  numbers  divisible  by  4  and  by  9.     What 
number  between  30   and  40  is  a  factor  of  each  of  them? 
Explain. 

139.  How  many  eights  in  1000  ? 

140.  To  1000  add  3  eights  or  24.     How  many  eights  in  the 
number  thus  formed  ? 


COMMON  DIVISORS  77 

141.  Add  to  1000  a  number  which  is  not  a  multiple  of  8. 
Is  the  sum  divisible  by  8  ?     Explain. 

PRINCIPLE  6.     A  number  is  divisible  by  8  if  the  number  ex- 
pressed by  its  three  right-hand  figures  is  divisible  by  8. 

142.  Tell  without  dividing  the  whole  number  which  of  the 
following  numbers  are  not  divisible  by  8,  and  give  remainders : 
3640.    5728.    9076.    4126.    5345.    1724.    8638.    1124.    10008. 

143.  Make  some  numbers  which  are  divisible  by  8  and  by  5, 
and  tell  how  you  make  them.     With  what  figure  do  they  end  ? 
What  two  multiples  of  10  besides  10  are  contained  in  each  of 
them? 

144.  Tell  how  to  compose  numbers  that  are  divisible  by  8, 
and  also  by  9  and  hence  by  72. 

COMMON  DIVISORS 

145.  A  number  which  is  a  factor  of  each  of  two  or  more 
numbers  is  called  their  Common  Divisor.     Illustrate. 

146.  Turn  to  the  diagram  on  page  67  and  find  all  the  num- 
bers in  it  of  which  11  is  a  common  divisor. 

147.  Name  all  the  numbers  less  than  100  of  which  8  is  a 
common  divisor.     Give  all  the  numbers  less  than  100  that 
have  as  a  common  divisor :  9.     10.     12.     6. 

148.  What  common  divisor  have  all  even  numbers  ? 

149.  Give  a  common  divisor  of  14,  21,  28,  35,  42,  49,  56,. 
and  63. 

150.  Name  three  multiples  of  11,  and  give  a  common  divisor 
of  them. 

151.  Make  a  list  of  sets  of  numbers  that  have  one  or  more 
common  divisors,  and  write  the  Greatest  Common  Divisor  of 
each  set. 


78  PROPERTIES  OF  NUMBERS 

SUGGESTION  FOR  CLASS  EXERCISE.  Let  a  pupil  name  two  or  more 
numbers  that  have  a  common  divisor,  and  let  the  class  discover  the 
divisor. 

152.  What  divisor  is  common  to  the  7th  even  number  and 
the  llth  odd  number  ? 

153.  2  is  a  common  divisor  of  10  and  of  20.     Is  it  the 
greatest  common  divisor  of  these  two  numbers  ? 

What  is  the  greatest  common  divisor  of  24  and  36  ? 

154.  Give  at  sight  the  greatest  common  divisor  of : 
a   10,  20,  and  40  g   70,  80,  and  90 
b    15,  30,  and  45  h  60,  72,  and  84 
c    18,  27,  and  45  i    63,  72,  and  90 

d   16  and  24  ./    28,  32,  40,  and  44 

e    50,  75,  and  100  k  15  and  25 

/    25,  30,  and  35  I    12,  18,  and  30 

155.  A  candy  manufacturer  filled  some  boxes  with  choco- 
lates, and  some  others  of  the  same  size  with  bonbons.     There 
were  24  Ib.  of  chocolates  and  28  Ib.  of  bonbons.     What  is 
the  largest  number  of  pounds  each  box  can  contain  ? 

156.  A  boy  wishes  to  divide  two  ropes,  one  42  ft.  long,  the 
other  56  ft.,  into  pieces  of  equal  length,  each  as  long  as  possible. 
How  long  will  each  piece  be  after  this  division,  and  how  many 
pieces  will  there  be  ? 

157.  Mr.  Allen  has  three  strips  of  land.     The  first  contains 
10  acres,  the  next  12  acres,  the  next  14  acres.     He  wishes  to 
lay  them  off  into  the  largest  possible  equal  lots.     How  many 
acres  will  there  be  in  each  lot,  and  into  how  many  lots  can 
each  piece  be  divided  ? 

158.  The  abbreviation  for  greatest  common  divisor  is  g.  c.  d. 
What  is  the  g.  c.  d.  of  35  and  65  ? 


POWERS  AND   ROOTS  79 

The  g.  c.  d.  of  two  numbers  may  be  easily  found  by  the  following 
process  of  continued  division.  It  is  to  be  used  with  numbers  which  are 
so  large  that  their  divisors  cannot  be  readily  found  by  inspection. 

159.  By  the  following  rule  find  the  g.  c.  d.  of  8  and  10 : 
To  find  the  greatest  common  divisor  of  two  numbers  — 
Divide  the  greater  number  by  the  less.     If  there  is  a  remainder, 

use  it  as  a  divisor  of  the  preceding  divisor,  and  continue  until  there 
is  no  remainder.    The  last  divisor  is  the  greatest  common  divisor. 

160.  Find  by  continued  division  the  g.  c.  d.  of  49  and  168. 

49)168(3  Using  49  as  a  divisor  of  168,  the  quotient  is  3  and 

147  remainder  21.    Using  21  as  a  divisor  of  49,  the  quo- 

21,)49(2  tient  is  2  and  remainder  7.     Using  7  as  a  divisor  of 

42  21,  the  division  is  exact,  hence  7  is  the  last  divisor,  or 

7)21(3  the  g.  c.  d.  of  49  and  168. 

161.  Find  by  continued  division  the  g.  c.  d.  of  the  following : 
a  24  and  132  i      77  and    847          q  198  and  252 

b  36  and  120  ./  18  and  243.  r  176  and  242 

c  35  and  105  k  96  and  224  s  361  and  431 

d  49  and  140  I  85  and  187  t  288  and  536 

e  64  and  480  m  125  and  175  u  84  and  154 

/  72  and  252  n  105  and  195  v  189  and  405 

g  30  and  735  o  135  and  245  w  960  and  204 

h  44  and  242  p  795  and  1105  x  236  and  576 

162.  CLASS  EXERCISE.     Let compose  two  large  num- 
bers having  a  common  divisor,  and  let  the  class  find  this 
common  divisor  by  continued  division. 

POWERS  AND  ROOTS 

163.  The  product  obtained  by  multiplying  a  number  by  itself 
one  or  more  times  is  called  a  Power  of  that  number.   Illustrate. 


80  PROPERTIES   OF   NUMBERS 

164.  The  product  of  two  equal  factors  is  the  Square  of  each 
factor.     2  x  2  is  expressed  22,  and  read  2  square,  or  2  to  the 
second  power.     Find  values  of  :  I2.    302.    502.     1202.    152.    202. 

165.  Give  quickly  in  order  the  first  12  numbers  that  are 
perfect  squares.     Learn  them. 

166.  92  equals  how  many  times  32?     162  how  many  times 
82  ?     82  equals  how  many  times  22  ? 

167.  What  two  perfect  squares  less  than  100  have  6  for 
their  units'  digit  ?     9  ?    4  ?    1  ? 

168.  Can  you  find  a  perfect  square  less  than  100  whose  tens' 
digit  is  9?    7?    5?    3? 

169.  How  much  is  TV  of  82?    Of  II2? 

170.  Multiply  62  by  the  first  prime  number  after  31. 

171.  The  product  of  three  equal  factors  is  called  the  Cube  of 
each  factor.     2  x  2  x  2  is  expressed  23,  and  read  2  cube,  or  2  to 
the  third  power.     Find  values  of  :  23.     33.     43.     53.     I3. 

172.  Continue  the  following  table  through  123.     Learn  the 
table.  I3  =  1 


173.  Name  a  perfect  cube  whose  units'  digit  is  :   1.     2.     3. 
4.     5.     6.     7.     8.     9.     0. 

174.  Multiply  the  cube  of  8  by  .07.     By  .125. 

175.  Find  6%  of  :43.     93.     73.     II3.     503.     123.     GO3. 

176     *=*          *=*          ^=9          *£=*          i*=? 

'      43  33  63  53  43 

SUGGESTION  TO  TEACHER.  By  the  following  work  lead  pupils  to  dis- 
cover the  relations  between  a  solid  that  is  a  cube  and  the  third  power  of  the 
number  that  measures  one  of  its  dimensions.  Inch  cubes  should  be  used 
in  this  work  until  pupils  are  able  to  image  the  solids  clearly  without  them. 


POWERS  AND   ROOTS 


81 


177.  How  many  cubic  inches  does  a  2-inch  cube  contain  ?     A 
3-inch   cube  ?     A   4-inch   cube  ?     A   5-inch  cube  ?     A   6-inch 
cube  ?     A  7-inch  cube  ?     An  8-inch  cube  ?     A  9-inch  cube  ? 

178.  One  of  the  boys  may  draw  a  square  yard  on  the  floor 
in  one  corner  of  the  room.     How  many  cubic  blocks  1  ft.  in 
dimensions  would  cover  the  square  yard  ? 

179.  One  of  the  girls  may  show  how  high  she  thinks  the 
blocks  must  be  piled  to  make  a  cubic  yard.     Another  member 
of  the  class  may  measure  with  the  yard  stick  and  see  how 
nearly  right  she  is. 

180.  How  many  cubic  feet  in  the  lowest  layer  of  blocks? 
How  many  layers  would  it  take  to  make  a  cubic  yard  ?  .  How 
many  cubic  feet  in  a  cubic  yard  ? 

181.  How  many  layers  of  inch  cubes  would  be  required  to 
cover  a  square  foot  ?     How  many  layers  of  the  inch  cubes  to 
make  a  cubic  foot  ?     How  many  inch  cubes  in  a  cubic  foot  ? 

182.  Into  how  many  2-inch  cubes   can   a  4-inch  cube   be 
divided  ? 

183.  How  many  6-inch  cubes  can   be  packed  into  a   box 

whose  inside  dimensions  are  each 
1ft.? 


6  Square  Centimeters 
FIG.  2. 


184.  Copy  Fig.  2  on  paper  or 
pasteboard,  making  each  square 
1  sq.  cm.  Cut  out  the  copy  and 
fold  and  fasten  it  so  that  it  will 
inclose  a  cubic  centimeter.  How 
many  such  cubes  would  be  re- 
quired to  cover  a  square  deci- 
meter? How  many  layers  of  them 


would  be  required  to  make  a  cubic  decimeter  ? 


185.    A  cubic  decimeter  is  called  a  Liter, 
centimeters  does  it  contain  ? 

HORN.    GRAM.    SCH.    AR.  6 


How  many  cubic 


82  PROPERTIES   OF  NUMBERS 

186.  Copy  Fig.   2   on   paper   or   pasteboard,  making   each 
square  1  sq.  dm.     Cut,  fold,  and  fasten  to  make  a  cubic  deci- 
meter or  a  liter. 

SUGGESTION  TO  TEACHER.  The  most  perfect  liters  and  cubic  centi- 
meters made  by  the  children  should  be  kept  as  a  part  of  the  school 
apparatus. 

187.  How  many  square  centimeters  in  all  the  surfaces  of  a 
liter  ? 

188.  In  the  metric  system  the  liter  is  the  measure  that  cor- 
responds  very   nearly   to   the   quart    in  the   English    liquid 
measure.     How  much  will  12  liters  of  oil  cost  at  65  cents 
per  liter? 

189.  Place  a  cubic  centimeter  upon  a  cube  that  holds  a  liter. 
How  many  cubic  centimeters  in  the  figure  thus  formed  ?   How 
many  square  centimeters  in  all  its  surfaces  ? 

SUGGESTION  TO  TEACHER.  It  should  be  explained  that  a  liter  is  a  unit 
of  measure,  and  not  a  fixed  form. 

190.  How  many  liters  will  be  contained  in  a  box  that  is  3 
dm.  long,  2  dm.  wide,  and  5  dm.  high? 

191.  About  how  many  liters  of  wheat  can  be  put  into  a  peck 
measure  ? 

192.  2  x  2  x  2  x  2  is  24,  which  is  read  2  to  the  fourth  power. 
Raise  to  the  fourth  power  each  of  the  first  five  numbers. 

193.  Kaise  to  the  fifth  power  each  of  the  first  three   odd 
numbers. 

194.  Raise  to  the  sixth  power  each  of  the  first  two  even 
numbers. 

195.  Raise  10  to  the  seventh  power. 

196.  Give  the  number  whose  prime  factors  are  2,  2,  and  3. 
2,  3,  and  3.    2,  5,  and  5. 


POWERS   AND   ROOTS  83 

197.  Find  values  of  x. 

a  x  =  32  x  5  e  x  =  2s  x    3  i  x  =  72  x  13 

6  ^  =  22  x  7  /  x  =  26  x    5  j  x  =  23  x    5   x  II2 

c  a;  =  23  x  34  g  x  =  33  x  11  A;  a;  =  2s  x    3   x    52 

d  »  =  24  x  33  h  x  =  53  x  11  J  a>  =  34  x    52  x    7 

198.  Each  of  the  two  equal  factors  that  compose  a  perfect 
square   is  called  a  Square   Root  of   that   number.     Give   the 
square  root  of  9.     25.     16. 

199.  V    is    used    as    the    sign  of    square  root.     Vl6  =  ? 
=  ?    V144=?    Vl21  =  ?    VI  =  ?    V6l=?    VlOO  =  ? 


200.  Divide  .63  by  Vl9.     By  V9.     By  V81. 

201.  Divide  .36  by  V9.     By  Vl6.     By  V4.     By  V8l. 

202.  Divide  5^44  by  V§1.    By  V36.    By  Vl21.    By  V49. 
By  V64.     By  Vl6.     By  V9.     By  Vl44. 

203.  How  long  is  one  side  of  a  square  whose  area  is  9  sq.ft.? 
49  sq.  ft.  ?     100  sq.  ft.  ?     81  sq.  ft.  ?     25  sq.  ft.  ? 

204.  How  much  is  50%  of  VlOO?     25%  of  V64?     75% 
ofV64?     25%ofVl44? 

205.  Multiply  Vl44  by  .3.     By  .05.     By  .007. 

206.  Find  6%  of  Vlll.     Of  V64.     Of  V8l. 

207.  How  much  is  3  times  V9?     4  times  V9? 

208.  "5V9"  is  read  "5  times  the  square  root  of  9."     Find 
the  value  of  the  expression. 


209. 

210.  3  V16  =  ?  4  VlOO  =  ?  2  V81  =  ?  6V64  =  ?  3  Vlll  =  ? 

211.  How  long   is   the   perimeter  of   a   square  containing 
49  sq.  in.  ? 


84  PROPERTIES   OF   NUMBERS 

212.  At  $  1.25  per  rod,  how  much  will  it  cost  to  fence  a 
square  lot  containing  25  sq.  rd.  ?     Eepresent. 

SUGGESTION  TO  TEACHER.  For  oral  "quick  work"  exercises  similar 
to  the  following  are  useful  :  "Think  of  the  3d  multiple  of  6,  subtract  2, 
take  the  square  root,  add  1,  square,  add  5,  take  ^,  take  j1^,  add  1,  square, 
add  5,  take  square  root."  Allow  children  to  lead  the  work,  letting  them 
prepare  their  numbers  beforehand  to  read  to  the  class,  until  they  are  able 
to  extemporize. 

213.  Each  of  the  three  equal  factors  that  compose  a  number 
that  is  a  perfect  cube  is  called  a  Cube  Root  of  that  number. 
Give  the  cube  root  of  :  8.    512.    64.    1000.    729.    1331.    1728. 
125.     216.     343. 

214.  Give  the  cube  root  of  a  perfect  cube  whose  units'  digit 
is:     1.     2.     3.     4.     5.     6.     7.     8.     9.     0. 

215.  How  long  is  one  side  of  a  cube  that  contains  8  cu.  in.  ? 
27cu.in.?    1728  cu.  in.?    64cu.in.?    729cu.in.?    1000  cu.  in.  ? 
512cu.in.?    216cu.in.?    125cu.in.?    343cu.in.? 

216.  ^/  is  read  "The  cube  root  of."     How  much  is  V64? 
•v/343?     -v/729?     -v/1728  ? 

217.  Multiply  the  -\/8  by  the  first  prime  number  after  40. 

218.  Multiply  -v/125  by  .001.     -^512  by  .75. 

219.  Find  50%  of  A/64.     Of  ^/1728. 

220.  -\/729--v/2l6  =  ?  -^1000  -^-  A/125  =?  A/1331  X  A/343  =? 


MISCELLANEOUS  EXERCISES 
1.    Write  in  decimal  form  and  add  :  1  ten-thousandth, 


1    hundred-thousandth,   y^fo,    TH^    1    millionth, 

1  00000* 

2.    If  you  have  a  string  a  foot  long  and  cut  one  inch  from 
each  end,  how  long  is  the  string  that  is  left? 


MISCELLANEOUS   EXERCISES  85 

3.  When  a  line  3.4  ft.  long  is  cut  from  each  end  of  a  line 
that  is  1  rd.  or  16.5  ft.  long,  how  long  is  the  line  that  is  left  ? 

4.  The  diagonal  of  a  certain  schoolroom  is  35.1  ft.      John 
makes  a  mark  on  the  diagonal  7  ft.  from  one  corner,  and  James 
makes  a  mark  9  ft.  from  the  opposite  corner.     If  each  boy 
stands  at  the  mark  he  has  drawn,  how  far  apart  are  they  ? 

SUGGESTION  TO  TEACHER.  In  most  classes  there  are  some  pupils  who 
fail  to  visualize.  Select  two  of  these  to  take  the  parts  of  John  and  James 
in  illustrating  this  and  similar  problems. 

5.  Find  the  sum  of  81.375  and  the  prime  number  nearest 
to  24. 

6.  Find  the  difference  between  21.84  and  the  largest  prime 
that  can  be  expressed  by  two  digits. 

7.  When  a  decimal  of  3 'places  is  multiplied  by  an  integer, 
how  many  decimal  places  should  be  pointed  off  in  the  product? 
Illustrate. 

8.  When  an  integer  is  multiplied  by  a  decimal  of  2  places, 
how  many  decimal  places  should  be  pointed  off  in  the  product  ? 
Illustrate. 

9.  How  many  decimal  places  in  the  square  of  .007  ? 

10.  8.283-*-  3  =  ?        45.6  --12=?        .286-^22=? 

11.  Tell  how  you  divide  a  decimal  by  a  decimal. 

12.  .12-j-.4=?     .15 -f-. 005=?     .75  -=-.5=?     .84 -.12=? 

13.  When  one  decimal  is  divided  by  another  decimal  of  the 
same  denomination,  how  many  decimal  places  are  there  in  the 
quotient  ? 

14.  At  $  .05  per  pound,  how  many  pounds  of  sugar  can  be 
bought  for  $  .45  ?     For  $  .75  ?  $  1.25  ?  $  2.50  ?  $  8  ? 

15.  Harriet  has  some  money  in  the  bank,  the  interest  of 
which  is  $  1.30  every  year.     How  long  must  the  money  stay  in 
the  bank  that  the  interest  may  be  $  5.85  ? 


86  .  PROPERTIES   OF  NUMBERS 

16.  1.728 -s-. 0012=? 

17.  Give  the  prime  factors  of  the  first  odd  composite  num- 
ber after  81. 

18.  The  largest  prime  factor  of  66  is  how  many  times  the 
smallest  prime  factor  of  66  ? 

19.  Find  the  1.  c.  m.  of  3,  8,  4,  9,  6,  12. 

20.  Find  the  g.  c.  d.  of  44  and  66.     Of  128  and  144. 

21.  Divide  7235.2  by  the  1.  c.  m.  of  4  and  7. 

22.  Divide  4.725  by  the  g.  c.  d.  of  45  and  105. 

23.  Find  by  cancellation  the  value  of  x : 


a 
7x8x4 

b 
3x7x9 

c 
64x21 

14x32 
d 
25x21 

21  x  18  x  5 
e 
48x63 

42  x  8  x  8 

/ 
16  x  25  x  36 

35x30 

g 

49  x  63 

36  x  24  x  18 
h 

48          x 

200  x  18  x  6 
i 
56 

21x84 

24x36 

21  x  16 

24.  What  is  one  of  the  two  equal  factors  of  121  ? 

25.  Name  a  perfect  square  whose  units'  digit  is  9,  and  give 
its  square  root. 

26.  Give  quickly  the  first  12  numbers  that  are  perfect  cubes. 

27.  How  long  is  one  edge  of  a  cube  that  contains  1000  cu. 
in.  ?     1728  cu.  in.  ? 

28.  Give  one  of  the  three  equal  factors  of  216.     Of  729. 

29.  Find  the  difference  between  O2  and  I2.     I2  and  22.     22 
and32.     32  and  42.    42  and  52.    52  and  62.    62  and  72.     72and82. 
82  and  92. 


MISCELLANEOUS   EXERCISES  87 

30.  Write  these  differences  in  a  column  and  tell  whether 
they  are  even  numbers  or  odd  numbers. 

31.  Find  the  sum  of  the  first  7  odd  numbers.     Compare  that 
sum  with  the  square  of  7. 

32.  Compare  the  sum  of  the  first  8  odd  numbers  with  the 
square  of  8.     The  sum  of  the  first  5  odd  numbers  with  the 
square  of  5.     Of  the  first  9  odd  numbers  with  the  square  of  9. 

33.  Find  the  sum  of  the  first  5  even  numbers.     Subtract  the 
square  of  5  from  that  sum. 

34.  Find  how  much  the  sum  of  the  first  7  even  numbers 
exceeds  the  square  of  7. 

35.  Take  Ex.  34,  substituting  other  numbers  for  7. 

36.  Ella's  record  on  an  arithmetic  test  was  75%.     What 
fractional  part  of  her  work  was  right  and  what  part  wrong  ? 

37.  Mr.  Hudson  had  $  8000  in  bank  and  took  out  20%  of  it. 
How  much  did  he  take  out  ?     How  much  had  he  left  ? 

38.  Edward  buys  oranges  at  the  rate  of  4  for  25  ^,  which  is 
just  one  half  of  what  he  receives  for  them.     What  is  the  selling 
price  of  each  ? 

39.  A  milkman's  horse  ran  away  with  a  wagon  containing 
4  gal.  of  milk,  and  25%  of  it  was  spilled.     How  many  quarts 
of  milk  were  spilled  ?     If  the  milk  was  worth  6  ^  per  quart, 
what  money  value  was  lost  ? 

40.  From  a  liter  of  oil  13%  was  spilled. 
How   many   cubic    centimeters   of    oil    re- 
mained ? 

41.  Draw  a  circle.     What  is  a  radius  ? 
Diameter  ?    Circumference  ?    The  radius  of 
a  circle  equals  what  part  of  the  diameter  ? 

42.  How  long  is  the  diameter  of  a  circle 
FlG-  3'              whose  radius  is  5  in.  ?     3J  in.  ?     7.5  in.  ? 


88 


PROPERTIES   OF   NUMBERS 


43.  What  is  the  diameter  of  the  largest  circle  that  can  be 
cut  from  a  piece  of  paper  3  in.  square  ? 

44.  The  surface  passed  over  in  1  hr.  by  the  minute  hand  of 
a  clock  is  what  figure  ?    The  minute  hand  of  a  clock  in  a  tower 
is  2-J-  ft.  long.     How  long  is  the  diameter  of  the  circle  it  passes 
over  every  hour  ? 

45.  A  plane  figure  bounded  by 
six  straight  lines  is  called  a  Hexa- 
gon.   When  the  sides  are  all  equal, 
and  the  angles  are  all  equal,  as  in 
Fig.  4,  the   hexagon   is   called   a 
Regular  Hexagon.      What  kind  of 
angles  has  a  regular  hexagon  ? 

46.  If   each  side  of   a  regular 
hexagon  is  6.75  in.  long,  how  long 
is  the  perimeter  of  the  hexagon  ? 


FIG.  4. 


47.  Draw  a  hexagon  that  is  not  regular. 

SUGGESTION  TO  TEACHER.  Show  the  following  method  of  drawing 
a  regular  hexagon  :  Draw  a  circle  with  a  radius  of  any  convenient 
length.  Beginning  at  any  point  of  the  circumference,  lay  off  the  radius 
as  a  chord  six  times  consecutively.  Erase  the  circle. 

48.  Draw  a  regular  hexagon  whose  sides  are  each  3  in.  long. 
How  long  is  the  perimeter  ?     How  dees  the  side  of  a  regular 
hexagon  compare  with  the  radius  of  the  circle  in  which  it  is 
inscribed  ? 

49.  By  drawing  diagonals  the  regular  hexagon  may  be  di- 
vided into  6  equilateral  triangles.     Draw  them,  and  find  how 
long  each  diagonal  is.     How  long  is  the  perimeter  of   each 
equilateral  triangle  ? 

50.  If  the  perimeter  of  the  hexagon  were  32.4  in.,  how  long 
would  the  perimeter  of  each  equilateral  triangle  be  ? 


51.    How  many  triangles  in  50%  of  the  hexagon  ? 


MISCELLANEOUS  EXERCISES 


89 


FIG.  5. 


52.    A  plane  figure  bounded  by  four  equal  straight  lines, 
and  having  no  right  angles,  is  called  a  Rhombus.     What  kind 
of  angles  has  a  rhombus  ? 

53.  Draw   a    rhombus   by   the    following 
method : 

Draw  the  line  AB  of  any  convenient  length. 
With  AB  as  a  base  construct  an  isosceles  triangle 
CAB,  making  AC  greater  than  f  of  AB.  With 
AB  as  a  base  construct  an  isosceles  triangle  ADB, 
making  AD  equal  to  AC.  Erase  the  construction 
line  AB.  (A  construction  line  is  a  line  forming  no 
part  of  a  figure,  but  used  simply  to  help  in  its  con- 
struction.) 

54.  Construct  a  rhombus  each  of  whose 
sides  is  5  in. 

55.  Mr.  Jones  laid  out  a  flower  bed  in  the 
shape  of  a  rhombus,  each  side  of  which  was 
4.75  ft.  long.     How  long  was  the  entire  edge 
of  the  flower  bed  ? 

56.  If  the  entire  edge  had  been  28.8  ft.  long,  how  long  would 
have  been  one  side  of  the  flower  bed  ? 

57.  Draw  a  rhombus  and  the  long  diagonal  of  the  rhombus. 
Into  what  kind  of  triangles  is  the  rhombus  divided  ? 

58.  If  the  side  of  a  rhombus  is  7.5  in.,  and  its  longer  diago- 
nal is  10.875  in.,  how  long  is  the  perimeter  of  one  of  the  tri- 
angles into  which  the  long  diagonal  cuts  the  rhombus  ? 

59.  Draw  a  circle  and  inscribe  a  hexagon. 
Join  the  vertex  of  each  alternate  angle  with 
the  center  of  the  circle.  Into  what  kind  of 
figures  is  the  hexagon  divided  ?  How  long 
would  the  perimeter  of  each  of  the  figures 
be,  if  the  radius  of  the  circle  were  8  cm.  ? 
FIG.  7.  12  cm.  ? 

60.    Each  rhombus  is  what  fractional  part  of  the  hexagon  ? 


90  PROPERTIES   OF   NUMBERS 

61.  Finish  the  following  course  of  reasoning: 
Since  the  whole  of  anything  equals  100%  of  it, 

.1  =  33J%  of  it. 
f  =  — %  of  it. 

62.  Write  in  each  rhombus  the  per  cent  which  it  is  of  the 
hexagon.     Shade  one  rhombus  and  tell  what  per  cent  of  the 
hexagon  is  unshaded. 

63.  How  much  is  331%  of  12?   21?   24?   30?   45?   48? 

64.  How  much  is  66f%  of  15?   27?   18?   36?   33?   6? 

65.  331%  of  a  school  of  48  pupils  are  boys.     How  many 
girls  are  there  ? 

66.  How  much  is  33^%  more  than  $  15  ?   $  300  ?    $  600  ? 

67.  How  much  is  66f  %  more  than  $  900  ?  $  1200  ?  $  1800  ? 

68.  $  3000  -  66f  %  of  3000  =  ?    2100  -  66f  %  of  2100  =  ? 

69.  Each  of  the  equal  sides  of  an  isosceles  triangle  is  33^% 
longer  than  the  base,  which  is  15  in.  long.     How  long  is  the 
perimeter  of  the  triangle  ? 

70.  A  merchant  found  that  some  of  his  goods  were  shopworn 
and  marked  them  at  a  reduction  of  25%  of  their  cost.     How 
were  goods  marked  that   cost   12^?     20^?     40^?     $1.00? 
$1.60?     $10.00? 

71.  Find  the  selling  price  of  goods  marked  at  the  following 
prices,  which  are  to  be  reduced  in  price  33^%  on  account  of 
being  out  of  style.    Cloaks  costing  $  7.50,  bead  trimming  $  1.50 
per  yard,  lace  ruffling  $  .57  per  yard. 

72.  A  grocer  bought  goods  at  the  following  prises.     For  how 
much  must  they  be  sold  to  gain  33 J%  ?     25%  ? 

a  Tomatoes  @  12^  per  pound.       d  Oranges  @  24^  a  dozen. 
b   Raisins  @  6^  per  pound.  e   Bananas  @  18^  a  dozen, 

c    Molasses  @  36^  a  gallon.  /  Potatoes  @  30^  a  bushel. 


MISCELLANEOUS  EXERCISES  91 

73.  At  the  end  of  a  season  a  merchant  decided  to  reduce 
prices  33J%  on  all  of  the  following  goods  whose  prices  were 
over  $1.00  and  to  reduce  them  66f  %  on  all  those  whose  prices 
were  less  than  a  dollar.     Find  the  new  selling  prices. 

a  Lace  @  $  1.80  per  yard.          d   Silk  @  $  2.70  per  yard. 
b    Ribbon  @  $  .75  per  yard.        e  Velvet  @  $  1.68  per  yard. 
c    Calico  @  $  .06  per  yard.         /  Alpaca  @  $  .60  per  yard. 

74.  William  is  15  years  old.     His  age  is  33|%  of  his  father's 
age.     How  old  is  his  father  ? 

75.  Mr.  Gage  had  $  396.66  in  a  bank  and  took  out  33^%  of 
it.     How  much  remained  in  the  bank  ? 

76.  Mrs.  Wallace  lent  Mr.  Brown  $  1200  until  the  interest 
amounted  to  66f  %  of  the  principal.     How  much  was  the  inter- 
est ?     How  much  did  Mr.  Brown  then  owe,  including  principal 
and  interest  ? 

77.  Write  the  following  fractions  in  a  column  and  opposite 
to  each  its  value  in  %  :     \.     \.     f .     \.     }.     f     f     f .     f . 

78.  How  many  minutes  in  33^%  of  an  hour?     In  66f%  ? 
25%?     20%?    40%?     50%? 

79.  How  many  hours  in  50%  of  the  time  from  9  A.M.  Mon- 
day to  9  A.M.  Tuesday  ?     In  331%  of  it  ?     In  75%  ?     20%  ? 

80.  How  many  square  centimeters  in  80%  of  a  square  deci- 

meter ?     In  25%  ?     In  331%  ? 

81.  Draw  the  equilateral  triangle 
ADC,  one  of  whose  sides  represents 
3  in.  With  DC  as  a  base  line  construct 
another  equilateral  triangle  ACB. 
Erase  AC.  What  kind  of  a  figure  is 
ABCD  ?  How  long  is  its  perimeter  ? 

82.  With  either  side  of  the  rhombus  as  a  base  line,  construct 
another  equilateral  triangle.  Erase  the  base  line.  What  kind 
of  a  figure  have  you  drawn  ?  How  long  is  its  perimeter  ? 


92 


PROPERTIES   OF   NUMBERS 


83.  Continue   adding   equilat- 
eral triangles  until  you  have  a 
regular  hexagon.     Complete  the 
following  reasoning. 

84.  Since  the  whole  of   any- 
thing      =  100%, 

iofit=    16|% 
fofit=-%£? 

FIG.  9.  85.    On  your   copyv  of  Fig.  9 

write  in  each  equilateral  triangle  the  %  which  it  is  of  the 
hexagon. 


86.    If  16f  %  of  the  hexagon  were  shaded,  what 
would  be  unshaded  ? 


of  it 


87.  What  %  of  the  hexagon  is  the  figure  AOCB?  AODCB? 
ABCDEFO?     CDEFAO?  ' 

88.  How  long  is  the  perimeter  of  the  six-pointed  star  repre- 

sented by  Fig.  10  if  each  side  is  3.5  in.  ? 
What  kind  of  angles  are  those  whose  ver- 
tices are  at  the  points  of  the  star  ? 

89.  Make  a  six-pointed  star. 

TJraw  a  regular  hexagon,  and  construct  an 
equilateral  triangle  upon  each  of  its  sides. 
Erase  the  sides  of  the  original  hexagon.  The 
star  may  also  be  made  by  prolonging  the  sides 
of  the  hexagon  until  they  meet. 

90.  Divide  your  star  into  6  equal  rhom- 
buses.    Write   in   each  rhombus  the   % 
which  it  is  of  the  star. 

91.  Put  a  letter  at  the  center  and  one 
at  each  angle  of  your  copy  of  Fig.  11  and 
tell  what  figure  is  33£%   of  it.     831%. 

FIG.  11.  50%.     66|%. 


MISCELLANEOUS  EXERCISES  93 

92.  What  is  16f%  of  12?     Of   24?     Of  72?     Of  84? 
Of  120?     Of  144? 

93.  What  is  83£%  of  18  ?  Of  30?  Of  48?  Of  66?  Of  144? 

94.  To  make  a  profit  of  16f  %  for  what  price  must  goods 
be  sold  that  cost  6^?     18^?     15^?     30^?     54^? 


95.  What  must  be  the  selling  price  of  the  same  goods  to 
allow  a  profit  of  831%  ? 

96.  Select  from  the  following  list  the  per  cents  which  are 
most  easily  used  by  reducing  them  to  common  fractions  in  their 
lowest  terms,  and  give  the  equivalent  fractions  : 

33J%       11%       16f%       25%       831%       50%       17% 
20%  3%       40%         75%         9%         66|% 

97.  James  had  a  dollar  and  lost  17  cents.     What  per  cent 
of  his  money  was  left  ? 

98.  Mary  has  only  a  dollar.     Can  she  lose  101%  of  it? 
Explain. 

99.  20  equals  what  part  of  30  ?     Express  it  in  per  cent. 


100.  CLASS  EXERCISE.         —  may  give  a  number,  and  the 
class  may  give  33|%  of  it.     16f  %.     66J%.     831%. 

101.  CLASS  EXERCISE.     -  may  give  a  number,  and  the 
class  may  give  the  number  of  which  his  number  is  16J%. 
331%.     25%. 

102.  Draw  a  right  triangle  whose  base  is  3  in.  and  perpen- 
dicular 4  in.     If  your  drawing  is  correct,  the  hypotenuse  will 
be  5  in.     Each  side,  of  the  triangle  equals  what  part  of  its 
perimeter  ? 

103.  Draw  a  right  triangle  whose  base  is  6  in.  and  perpen- 
dicular 8  in.    Its  hypotenuse  is  just  twice  as  long  as  the  hypot- 
enuse of  the  triangle  given  in  Ex.  102.     Each  of  its  sides 
equals  what  part  of  its  perimeter  ? 


94  PROPERTIES  OF  NUMBERS 

104.  Draw  a  square  3  in.  in  dimensions.     If  you  drew  a 
larger  square,  having  each  of  its  sides  1  in.  from  the  corre- 
sponding side  of  the  first  square,  how  long  would  its  per- 
imeter be? 

105.  Separate    the   following    into   two   lists,   one   of  odd 
numbers,  the  other  of  even  numbers.     How  many  are  there  of 
each  ? 

874;  MDCCCLXXXVIII ;  the  square  of  7 ;  the  fifth  multiple 
of  4 ;  the  product  of  7  and  8 ;  the  quotient  of  84  divided  by  2 ; 
the  difference  between  81  and  18 ;  the  sum  of  85  and  37 ;  the 
largest  numbe*r  that  can  be  expressed  by  two  figures ;  the 
largest  factor  of  12  except  itself ;  the  number  that  is  5  greater 
than  212 ;  the  largest  number  that  can  be  expressed  by  three 
figures ;  the  smallest  number  that  can  be  expressed  by  three 
figures ;  the  number  that  means  a  dozen ;  the  number  that  tells 
how  many  days  in  May;  the  integer  between  17,345  and 
17,347 ;  one  of  the  equal  factors  of  25 ;  the  factor  that  helps  7 
to  make  77 ;  the  square  root  of  100 ;  the  number  that  shows 
how  many  quarts  in  a  peck;  the  denominator  of  the  fraction 
T\;  the  greatest  common  divisor  of  6  and  8 ;  the  remainder 
after  dividing  25  by  11 ;  the  smallest  multiple  of  7  that  will 
contain  5 ;  the  least  common  multiple  of  4  and  7 ;  the  number 
that  is  just  half  way  between  30  and  50;  the  smallest  prime 
number  greater  than  25 ;  the  largest  prime  number  less  than 
25;  the  numerator  of  the  fraction  |^;  the  number  that  tells 
how  many  square  inches  in  a  square  foot ;  the  number  that  is 
just  as  much  less  than  15  as  it  is  greater  than  11;  the  average 
of  19,  20,  and  21 ;  the  first  composite  number ;  the  number 
that  shows  how  many  pounds  in  a  ton ;  the  number  that  shows 
how  many  cubic  inches  in  a  cubic  foot ;  the  number  that  shows 
how  many  sides  a  pentagon  has ;  the  largest  prime  number  that 
can  be  written  with  two  figures ;  the  smallest  prime  number 
that  can  be  written  with  three  figures ;  the  quotient  of  13.14 
divided  by  .06;  50%  of  862;  the  largest  prime  factor  of  102  j 
the  number  that  shows  how  many  millimeters  in  a  meter. 


CHAPTER   III 

RATIO 

SUGGESTION  TO  TEACHER.  Review  ratio  as  given  in  Hornbrook's 
"Primary  Arithmetic."  See  notes  on  pp.  117  and  118  and  tables  on 
pp.  145,  160,  174,  183,  of  that  book. 

1.  A  3-inch  line  equals  what  part  of  a  4-inch  line,  or  what 
is  the  ratio  of  a  3-inch  line  to  a  4-inch  line  ? 

2.  What  is  the  ratio  of  a  pint  to  a  quart  ?     Of  a  quart  to  a 
gallon  ?     Of  an  inch  to  a  foot  ?    Of  a  foot  to  a  yard  ?    Of  2  ft. 
to  a  yard  ?     Of  an  ounce  to  a  pound  ?     Of  8  oz.  to  a  pound  ? 
Of  15  oz.  to  a  pound  ? 

3.  CLASS  EXERCISE.     may  name  a  number  less  than 

100,  and  the  class  may  give  its  ratio  to  100. 

4.  6  is  how  many  times  3,  or  what  is  the  ratio  of  6  to  3  ? 

5.  What  is  the  ratio  of  a  yard  to  a  foot?     Of  a  foot  to  an 
inch  ?     Of  a  foot  to  3  in.  ?     Of  a  foot  to  6  in.  ?     Of  a  foot  to 
7  in.  ?     9  in.  ?     11  in.  ? 

6.  CLASS  EXERCISE.     may  name  some  number  greater 

than  10,  and  the  class  may  give  its  ratio  to  10. 

7.  The  ratio  of  two  numbers  is  the  quotient  of  the  first  of 
those  numbers  divided  by  the  second.     Thus  the  ratio  of  10  to 
5  is  10  -f-  5,  or  2.     The  ratio  of  7  to  5  is  7  -^  5,  or  If.     What 
is  the  ratio  of  4  to  5  ? 

8.  Draw  a  rectangle  4  in.  long  and  1  in.  wide.     A  rectangle 
3  in.  long  and  1  in.  wide  equals  how  many  fourths  of  the  first 
rectangle  ?     A  rectangle  8  in.  long  and  1  in.  wide  equals  how 
many  fourths  of  the  first  rectangle  ?     What  do  j  equal  ? 

95 


96 


RATIO 


The  ratios  indicated  by  "  parts  "  and  "  times  "  are  really  of  the  same 
kind.  They  both  express  the  quotient  of  one  quantity  divided  by  another 
of  the  same  kind. 

9.    What  is  the  ratio  of  a  second  to  a  minute  ?     Of  a  year 
to  a  month  ? 

10.  Build  from  inch  cubes  or  draw  right  prisms  like  the 
following : 


3x2x1 


5x2x1 


7  /  / 


2x2x2 


3x3x1 

arn 

IIT 


I 

fir 


5x1x1 


7x2x1 


I 


1 

1 

1 1 .    Find  ratios  of : 
a  to  b  a  to  c 

b  to  c  b  to  d 

c  to  b  c  to  d 


FIG.  1. 

&  to  e 
e  to  d 
btof 


ftod 
c  to/ 
c  to  e 


d  to  c 
/to  a 
a  to  e 


12.  Mr.  Jones  works  every  day  from  8  until  12  o'clock,  and 
from  1  until  5  o'clock.  At  9  o'clock  in  the  morning,  what  is 
the  ratio  of  the  work  he  has  done  to  the  work  he  still  has  to 


RATIO  97 

do  that  day  ?    What  is  the  ratio  of  the  work  he  has  done  to 
his  whole  day's  work  ? 

13.  At  ten  o'clock,  what  is  the  ratio  of  the  work  he  has  done 
to  a  day's  work?    At  12  o'clock?    At  1  o'clock?    At  3  o'clock? 
At  5  o'clock  ? 

14.  What  is  the  ratio  of  a  rod  to  a  mile  ? 

15.  If  your  home  is  a  mile  from  the  schoolhouse,  how  many 
rods  must  you  travel  each  school  day  of  two  sessions,  if  you 
go  home  at  noon  ? 

16.  Joseph  rode   a  mile   on  his   bicycle.     When  he  had 
ridden  a  rod,  what  was  the  ratio  of  the  distance  he  had  ridden 
to  that  which  he  afterward  rode  ? 

17.  Ella  walked  to  the  home  of  her  cousin,  who  lived  a 
mile  away.    What  was  the  ratio  of  the  distance  she  had  walked 
to  the  remaining  distance  after  she  had  gone  16  rd.  ?    32  rd.  ? 
80  rd.  ?    120  rd.  ? 

18.  What  is  the  ratio  of  an  ounce  to  a  pound  ? 

19.  Margaret  had  half  a  pound  of  candy  and  gave  away  all 
of  it  except  one  ounce.     What  was  the  ratio  of  what  she  had 
left  to  what  she  had  at  first  ? 

20.  What  is  the  ratio  of  a  pound  to  a  ton  ? 

21.  Just  after  a  ton  of  hay  was  weighed  in  market,  a  horse 
ate  one  pound  of  it.     What  was  the  ratio  of  what  he  ate  to 
what  was  left  ? 

22.  Ratio  is  expressed  by  a  colon.     Give  ratios  of :  15 : 3. 
3:15.  16:2.  2:16.  3:18.  18:3.  5:20. 

23.  Give  quickly  the  ratio  of  2  to  each  of  the  first  ten  mul- 
tiples of  2.     Give  the  ratio  of  the  2d  multiple  of  2  to  each  of 
the  first  ten  multiples  of  2.     Do  the  same  with  the  3d  multiple 
of  2.     With  the  4th,  5th,  6th,  7th,  8th,  9th,  and  10th. 

110KN.    GRAM.    SCH.    AR. — 7 


98  RATIO 

24.  What  is  the  ratio  of  the  2d  multiple  of  any  number  to 
its  3d  multiple  ?     Of  its  2d  multiple  to  its  4th  ?    Illustrate. 

25.  What  is  the  ratio  of  2471  to  17  ? 

26.  What  is  the  ratio  of  1.422  to  1.8  ? 

27.  The  ratio  of  3  to  6  is  -J ;  the  ratio  of  6  to  3  is  2.     These 
two  ratios  between  the  numbers  3  and  6  are  called  reciprocal 
ratios.     Give  the  reciprocal  ratios  between  the  following  num- 
bers :  2  and  3.     3  and  5.     8  and  4.     9  and  12.     18  and  20. 

28.  19  equals  how  many  twentieths  of  20  ?     20  equals  how 
many  nineteenths  of  19  ? 

29.  John  is  8  yr.  old,  and  his  sister  Mary  is  16  yr.  old. 
What  is  the  ratio  of  Mary's  age  to  John's  ?     Of  John's  age  to 
Mary's  age  ? 

30.  When  is  the  ratio  of  one  number  to  another  number  an 
integer  ?     Illustrate. 

31.    The  line  A B  rep- 

ACT)  JP  f'  B 

T I I I I 1      resents    a   distance    of 

Fl«-  2-  35  mi.  divided   into  5 

equal  parts.   How  much  is  the  distance  AE  ?    CB  ?  AF  ?   EB  ? 

32.  Find  the  ratio  of  AF  to  AB.     CF  to  CB.    AB  to  AD. 
AB  to  DB.     AD  to  CF.     AB  to  CF. 

33.  What  is  the  ratio  of  the  first  composite  number  after  19 
to  the  first  composite  number  after  29  ?     Of  the  first  odd 
number  after  20  to  the  first  odd  number  after  5  ?     Of  the  first 
prime  number  after  7  to  the  first  even  number  after  20  ? 

34.  Draw  a  2-inch  square  and  a  4-inch  square  and  find  the 
ratio  of  each  square  to  the  other. 

35.  Find   the  reciprocal   ratios  of  a  3-inch   square  and  a 
4-inch  square.     Of  an  8-inch  square  and  a  6-inch  square. 


RATIO  99 

36.  What  is  the  ratio  of  3  V4  to  2  V25  ?     2V9:6Vl6  =  ? 

37.  2V49:3Vl21  =  ?  3V36  :  2  VI44  =  ? 

38.  5Vl6:4V9  =  ?  2  V81  :  3  V64  =  ? 

39.  3  VlOO  :  5  Vl6  =  ?  2\/25:5V36  =  ? 

40.  Build  with  cubes  and  find  the  following  ratios  :     An 
inch  cube  to  a  3-inch  cube.     An  inch  cube  to  a  4-inch  cube.     A 
2-inch  cube  to  a  3-inch  cube. 

41.  If  a  5-inch  cube  is  cut  into  inch  cubes,  what  is  the  ratio 
of  one  of  the  small  cubes  to  the  large  cube  ? 

42.  What  is  the  ratio  of  a  cube  an  inch  in  dimensions  to  a 
6-inch  cube  ? 

43.  33:43=?     403:603  =  ?     113:93=?     63:203=? 

44.  Give  quickly  the  cube  root  of  :    27.    64.    8.    216.    512. 
729.     343.     1000.     125. 

45.  A/125  :  ^/lOOO  =  ?   ^/729  :  \/2l6  =  ?   -\/1728  :  -J/512  =  ? 


46.  Image  the  following  figures  and  tell  the  ratio  of  one  side 
of  each  figure  to  its  perimeter.    A  square.    A  regular  hexagon. 
A  rhombus.     A  regular  pentagon.     Express  the  ratios  in  °/0. 

47.  What  is  the  ratio  of  the  perimeter  of  a  square  yard  to 
the  perimeter  of  a  square  foot  ?     Of  the  perimeter  of  a  square 
centimeter  to  the  perimeter  of  a  square  decimeter  ? 

48.  What  number  is  that  whose  ratio  to  8  is  f  ?     Or  what 
is  I  of  8  ? 

49.  How  much  is  |  of  12  ?     f  of  49  ?     f  of  15  ?     \  of  21  ? 

SUGGESTION  TO  TEACHER.  In  order  to  insure  correct  reasoning  on  the 
part  of  pupils,  they  should  occasionally  be  required  to  explain  the  steps 
by  which  they  arrive  at  results,  as  :  since  \  of  21  is  3,  f  of  21  are  4  times 
3  or  12.  After  this  is  thoroughly  understood,  the  habit  of  mental  cancel- 
lation should  be  encouraged.  For  instance,  in  finding  ^  of  21,  children 
may  be  led  to  visualize  the  expression  and  mentally  to  cancel  the  terms. 


100  RATIO 

| 

50.  Find  values : 

|  of  36      |  of  56      £of63      ^  of  77      J  of  72      f  of  36 

51.  If  a  boy  earns  $  77  in  11  wk.,  how  much  would  he  earn 
in  3  wk.  ?     5  wk.  ?     9  wk.  ? 

52.  When  9  yd.  of  calico  cost  72^,  what  is  the  cost  of  2  yd.  ? 

5  yd.  ?     7  yd.  ?     8  yd.  ? 

53.  Goods  that  cost  8^  a  yd.  are  sold  for  f  of  their  cost. 
What  is  the  selling  price  ? 

54.  Give  quickly  the  selling  price  of  goods : 
a    Bought  at  $  0.12  and  sold  at  f  of  the  cost. 

b  Bought  at  f  0.18  and  sold  at  £  more  than  cost, 

c  Bought  at  $  0. 20  and  sold  at  \  more  than  cost. 

d  Bought  at  $  0.40  and  sold  at  ^  more  than  cost. 

e  Bought  at  $0.50  and  sold  at  £  more  than  cost. 

/  Bought  at  $  0.60  and  sold  at  ^  more  than  cost. 

g  Bought  at  $  0.80  and  sold  at  \  less  than  cost. 

h  Bought  at  $  1.00  and  sold  at  f  of  the  cost. 

i  Bought  at  $  0.40  and  sold  at  f  of  the  cost. 

SUGGESTION  TO  TEACHER.  Before  the  following  work  is  taken  up, 
pupils  should  be  drilled  in  finding  reciprocal  ratios  of  pairs  of  numbers 
until  they  readily  see  the  truth  of  the  first  statement  in  the  solution  of 
Ex.  55. 

55.  6  is  f  of  what  number  ? 

SOLUTION.     6  is  f  of  the  number  that  is  f  of  6.     %  of  6  is  2.     f  of 

6  are  8. 

56.  8  is  |  of?     f  of?     £  of?     I  of? 

57.  12  is  |  of?     *  of?     fof? 

58.  Find  the  values  of  x. 

a    10  =  §  of  x.     f  of  x.     |  of  x.     f  of  a.     |  of  x. 

b  12  =  |  of  x.  |  of  x.  f  of  x.  |-  of  x.  f  of  x.  T2T  of  x. 

c  15  =  |  of  x.  f  of  x.  f  of  x.  f  of  x.  |  of  x.  f  of  x. 


KATIO  101 

d  18  =  f  of  ar.  T%  of  x.  T6T  of  ».  -&•  of  x.  f  of  a.  f  of  a?, 
e  20  =  J  of  a;.  ^  of  x.  if  of  aj.  -V°-  of  x-  V  of  ^  V  of  aj- 
/  5  =  |  of  a;.  10  =  fofx.  7  =  |ofa?.  9  =  $  of  a;. 

59.  Anna's  age  is  f  of  Mary's  age.     What  is  the  ratio  of 
Mary's  age  to  Anna's  age.     If  Anna  is  12  years  old,  how  old 
is  Mary  ? 

60.  James  has  24  marbles.     He  has  %  as  many  as  John. 
How  many  marbles  has  John  ? 

61.  Land  in  one  part  of  a  certain  county  in  Illinois  is  worth 
$20  an  acre,  which  is  only  £  of  the  price  of  land  in  another 
part  of  the  county.     What  is  the  price  of  the  better  land? 
What  is  the  value  of  the  land  owned  by  Mr.  Baxter,  who  has 
40  acres  of  each  kind  ? 

62.  Mr.  Walker  sold  gingham  at  8^  a  yard,  which  was  f  of 
what  it  cost  him.     How  much  did  it  cost  ? 

3.  Find  the  cost  of  goods : 
a  Sold  at  $  0.09,  which  was  f  of  the  cost. 
b  Sold  at  $0.20,  which  was  f  of  the  cost, 
c  Sold  at  $  1.98,  which  was  f  of  the  cost. 
d  Sold  at  $  2.97,  which  was  f  of  the  cost. 
e  Sold  at  $  1.47,  which  was  -J  of  the  cost. 

64.  24  marbles  will  cost  how  many  times  as  much  as  12 
marbles  of  the  same  kind  ? 

65.  What  is  the  cost  of  24  marbles  when  12  marbles  cost 
$.25?     $.08?     $.60? 

66.  What  is  the  ratio  of  the  price  of  10  hats  to  the  price  of 
1  hat  ?     To  the  price  of  2  hats  ?     5  hats  ?     7  hats  ? 

67.  What  will  be  the  cost  of  10  hats  when  5  hats  cost  $3  ? 
$7?     $9?     $7.50? 

Use  ratio.     10  hats  cost  how  many  times  as  much  as  5  hats  ? 


102  RATIO 

68.  What  will  be  the  cost  of  10  hats  when  2  hats  are  worth 
$3?     $4?     $5.25?     $7.65? 

69.  If  3  articles  of  the  same  kind  cost  $.17,  how  much  will 
12  such  articles  cost  ?    18  articles  ?    21  articles  ?    30  articles  ? 

70.  If  5  things  cost  $19,  how  much  will  15  things  of  the 
same  kind  cost  ? 

71.  Take  Ex.  70,  substituting  another  number  for  5  and  for 
15  some  multiple  of  the  number  that  you  have  substituted. 

72.  If  10  acres  of  land  are  sold  for  $375,  how  much  would 
80  acres  cost  at  the  same  rate  ?    60  acres  ?     100  acres  ? 

73.  Find  the  cost  of  48  oranges  when  5^  are  paid  for  6 
oranges.     For  8  oranges.     For  4  oranges.     For  12  oranges. 

74.  If  15  marbles  are  sold  for  9^,  how  much  do  5  marbles 
cost  ?    3  marbles  ?    20  marbles  ? 

MISCELLANEOUS  EXERCISES 

1.  Add  the  square  of  7.9  to  7.9. 

2.  Subtract  the  cube  of  1.3  from  10. 

3.  Divide  142  by  .007. 

4.  A  regular  hexagon  is  inscribed  in  a  circle  whose  radius 
is  8  in.    How  long  is  the  perimeter  of  the  hexagon  ?    Eepresent. 

5.  How  long  is  one  side  of  a  regular  hexagon  whose  perim- 
eter is  5.4  cm.  ? 

6.  How  long  is  one  side  of  a  regular  pentagon  whose  perim- 
eter is  8.45  cm.  ? 

7.  Is  it  correct  to  say  that  an  inch  line  is  \  of  a  4-inch  line  ? 
If  the  inch  line  were  in  Boston  and  the  4-inch  line  in  New  York, 
would  the  shorter  line  be  a  part  of  the  longer  ?     What  part  of 
the  longer  line  would  the  shorter  line  equal  ? 


MISCELLANEOUS   EXERCISES  103 

8.  What  is  the  ratio  of  an  hour  to  a  day  ?     Of  a  week  to  a 
day? 

9.  Give  the  ratio  of  21  to  each  of  the  first  12  multiples 
of  7. 

10.  What  is  the  ratio  of  a  square  whose  side  is  3  ft.  to  a 
rectangle  9  ft.  by  8  ft.  ? 

11.  Find  the  prime  factors  of  546.     Of  495. 

12.  Find  the  ratio  of  the  largest  prime  factor  of  35  to  the 
largest  prime  factor  of  39.     Of  the  largest  prime  factor  of  49 
to  the  largest  prime  factor  of  15.     Of  the  smallest  prime  factor 
of  49  to  the  smallest  prime  factor  of  15. 

13.  603  :  303  =  ?     63 :  53  =  ?         73 :  53  =  ?     83  :  123  =  ? 

14.  7V4:3V49  =  ?  4V25  :  2 VTOO  =  ? 

15.  How  long  is  the  shortest  line  that  can  be  divided  into 
either  8-inch  lines  or  10-inch  lines  ? 

16.  Divide  the  1.  c.  m.  of  2  and  5  by  the  1.  c.  in.  of  3  and  5. 

17.  Divide  the  g.  c.  d.  of  36  and  45  by  the  g.  c.  d.  of  12  and  3. 

18.  Which  power  of  6  is  216  ? 

19.  Which  power  of  2  is  16  ?     64?    256? 

20.  Which  power  of  10  is  the  denominator  of  the  decimal 
.11?     .0125?     .6?     .345?     .00^004?     .000009? 

21.   AB  represents  a  distance  of  320  rd. 

J- 1 — £      or  1  mi.,  which  is  just  f  of  the  distance 

„      „  from  A  to  G.     How  far  is  it  from  A  to  C  ? 

r  IG.  o. 

From  B  to  C? 

22.  Mary  had  some  money  in  a  toy  bank.     She  took  out  35^ 
which  was  |-  of  it.     How  much  was  left  ? 

23.  Lizzie  spent  15^  for  a  singing  book,  which  was  J  of  the 
price  of  her  arithmetic.     The  price  of  the  arithmetic  was  -f  of 
the  price  of  her  story  book.     How  much  did  they  all  cost  ? 


104  RATIO 

24.  On  Monday  John  rode  on  his  bicycle  21  mi.,  which  was 
f  of  the  distance  he  rode  during  the  rest  of  the  week.     How 
far  did  he  'ride  during  the  whole  week  ? 

25.  A  boy  had  12  agates,  for  which  he  paid  60  £     He  sold 
them  for  6^  apiece.     How  much  did  he  gain  on  each  ? 

26.  Mr.  Cooper  paid  $64,  which  was  50%  of  what  he  owed. 
How  much  does  he  still  owe  ? 

27.  When  a  man  hires  a  house  to  live  in,  he  is  said  to  pay 
rent  for  it.     When  he  hires  money  to  use,  he  is  said  to  pay  in- 
terest for  it.     If  you  have  deposited  $  100  in  a  bank  that  pays 
4%  interest,  how  much  interest  will  you  receive  each  year? 

28.  How  much  would  you  receive  each  year  if  you  had  $  100 
at6%?     3%?     7%? 

29.  If  a  man  borrows  $25  and  pays  .06  of  $25  as  interest 
for  1  yr.,  how  much  interest  does  he  pay  ? 

30.  At  6%  what   is  the   interest   of  $14  for  a  yr.  ?     Of 
$16?     $30?     $40?    $60? 

31.  At  8%  what  is  the  interest  each  year  of  $  7  ?    $  11  ? 

32.  At  1%  what  is  the  interest  each  year  of  $  9  ?    $12  ? 

33.  At  8%  what  is  the  interest  of  $12  for  a  year?     For 
2yr.?     6yr.?     8  yr.  ?     12  yr.? 

34.  At  8%  what  is  the  interest  of  $  7  for  a  year  ?   For  |  yr.  ? 
Forl^yr.?     2£  yr.  ?     3£  yr.  ?     1J  yr.  ?     3J-  yr.  ?     1\  yr.  ? 

35.  At  6%  what  is  the  interest  of  $8  for  a  yr.  ?    For  \  yr.  ? 
For3£yr.?     4£yr.?     5£  yr.  ?     6J  yr.  ? 

36.  CLASS  EXERCISE.     may  tell  how  many  dollars  he 

would  like  to  have  at  interest  at  6%,  and  the  class  may  tell 
how  much  interest  he  would  have  each  year  from  it.     How 
much  in  2  yr.     3  yr.     4|-  yr.     5£  yr.     6£  yr. 


MISCELLANEOUS   EXERCISES  105 

37.  Mrs.  Ware  lent  $800  @  4%  interest,  $900  @  3%,  and 
$2500  @  5%.     How  much  interest  did  she  receive  each  year 
from  those  loans  ? 

38.  What  is  the  ratio  of  a  5-inch  square  to  an  oblong  13  in. 
long  and  5  in.  wide  ? 

39.  Draw  on  paper  a  rectangle  4  in.  long  and  3  in.  wide. 
Draw  a  diagonal.     Cut  out  the  rectangle  and  divide  it  along 
the   diagonal.      Into  what  kind   of  triangles   is  a  rectangle 
divided  by  a  diagonal  ?     What  is  the  area  of  the  rectangle  ? 
Of  each  triangle  ? 

40.  Show  the  truth  of  the  following  statement : 

The  area  of  a  right  triangle  equals  one  half  the  area  of  a 
rectangle  which  has  the  same  base  and  altitude. 

41.  What  is  the  area  of  a  right  triangle  whose  base  is  8  in. 
and  altitude  5  in.  ? 

42.  Give  directions  for  finding  the  area  of  a  right  triangle. 

43.  Find  the  area  of  a  right  triangle  whose  base  is  8  cm. 
and  altitude  9  cm.     Base  27  in.,  altitude  13  in. 

44.  Find  the  area  of  a  right  triangle  whose  base  is  12  in. 
and  whose  altitude  is  25%  of  the  base. 

45.  Can  you  bisect  a  rectangle  and  place  the  two  parts  so  as 
to  form  an  isosceles  triangle  ?     Kepresent. 

46.  Bisect  a  rectangle  and  place  the 
parts  as  in  Fig.  4.  Show  two  horizontal 
parallel  lines ;  two  oblique  parallel  lines. 

47.    A    four-sided    plane    figure    whose 
opposite    sides    are     parallel,    and    whose 
angles   are    not   right   angles,   is   called   a 
FIG.  4.         *  Rhomboid.     Draw  a  rhomboid. 


106 


RATIO 


48.  How  long  is  the  perimeter 
of  a  rhomboid  whose  short  sides  are 
each  7  in.  and  long  sides  each  10  in.  ? 


FIG.  5. 


49.    How  long  is  the  perimeter  of 
a  rhomboid  two  of  whose  sides  are 
each  1.75  ft.  long,  and  the  other  two  each  2.5  ft.  long  ? 


50.  Represent  and  find  perimeters  of  rhomboids  having : 
A  long  side,  18  in. ;  short  side,  33^%  of  a  long  side. 

A  short  side,  12  in. ;  long  side,  25%  longer  than  a  short  side. 

A  short  side,  1  ft.  4  in. ;  long  side,  50%  longer  than  a  short 
side. 

51.  What  per  cent  of  the  angles  of  a  rhomboid  are  obtuse  ? 

52.  Find  a  rhombus  in  your  book,  and  see  whether  it  agrees 
with  the  definition  of  a  rhomboid. 

A  rhombus  differs  from  other  rhomboids  in  having  all  its  sides  equal. 

53.  Draw  and  cut  out  two  equal 
equilateral  triangles.     Cut  them  in 
two,  and  arrange  the  right  triangles 
thus  formed  as  in  Fig.  6.    How  long 
would  the  perimeter  of  Fig.  6  be  if 
each  side  of  the  equilateral  triangles 
were  6  cm.  long?    16  cm.?    25  cm.? 

54.  Place  the  four  triangles  as  in 
Fig.  7.     What  kind  of   a  figure  is 
formed  ?     How  long  is  its  perimeter 
if  each  side  of  the  original  triangles 
is  10  in.  long  ?     11  in.  long  ? 

55.  By  changing  the  position  of 
one  triangle,  change  the  figure  into 
a  rhomboid.     Find  length  of  perim- 


FIG.  6. 


FIG.  7. 
eter  if  each  side  of  the  original  triangles  is  7  in.  long. 


MISCELLANEOUS  EXERCISES 


107 


56.  Place  the  four  triangles  as  in 
Fig.  8,  and  name  the  figure.  How 
long  would  the  perimeter  of  Fig.  8 
be  if  each  side  of  the  two  equilateral 
triangles  were  6  in.  long?  1  ft.  3 

FIG.  8.  in-  lon§? 

57.    Place  the  four  triangles  so  as  to  form  a  rectangle. 

58.  Draw  a  2-inch  square.  Draw  its 
diagonals.  With  the  point  where  the 
diagonals  meet  as  a  center  and  a  radius 
of  1  in.,  draw  a  circle.  At  how  many 
points  do  the  sides  of  the  square  touch 
the  circle  ?  At  how  many  points  do  the 
diagonals  cut  the  circumference  ?  Draw 
lines  AB,  BC,  etc.,  between  these  points 
as  in  Fig.  9. 

59.  Erase   all   but   the  part   shown   in 
Fig.  10.     Such  a  figure  is  called  an  Octagon. 
How  many  sides  has  an  octagon?      How 
long  would  the  perimeter  of  your  octagon 
be  if  each  side  were  7  in.  ?     9  in.  ? 

60.  Fold  the  octagon  in  various  ways, 
and  see  whether  the  angles  are  all  equal, 
and  whether  the  sides  are   all   equal.     If 

they  are,  what  kind  of  an  octagon  is  it  ? 

61.    What  kind  of  angles  has  a  regular  octagon  ? 

62.  Draw  diagonals  of  the  octagon  as  in 
Fig.  11.     Into  how  many  isosceles  triangles 
is  the  octagon  divided?    Each  triangle  is 
what  part  of  the  octagon  ? 

63.  If  the  area  of  the  octagon  were  60.48 
sq.  in.,  what  would  be  the  area  of  one  of  the 

FIG.  11.  isosceles  triangles  ? 


FIG.  10. 


CHAPTER   IV 

FRACTIONS 

1.  Draw  a  line  an  inch  long  and  divide  it  into  halves  and 
quarters.     How  many  halves  of  an  inch  are  there  in  an  inch  ? 
How  many  fourths  of  an  inch  ?     How  many  eighths  ?     How 
many  thousandths  of  an  inch  ?     How  many  millionths  ? 

A  Fraction  is  an  expression  of  one  or  more  of  the  equal 
parts  into  which  a  unit  is  divided. 

2.  In  the  expression  J,  4,  the  denominator  of  the  fraction, 
shows  that  some  unit  is  considered  as  separated  into  4  equal 
parts ;  3,  the  numerator,  shows  how  many  of  those  parts  are 
expressed.     What  is  meant  by  the  expression  £  of  an  inch  ? 

3.  In  the  fraction  |f  which  number  is  the  denominator? 
What  does  the  16  show  ?     What  name  is  given  to  the  number 
above  the  line  ? 

4.  What  is  meant  by  |-  of  an  inch  ?    |  of  an  apple  ? 

5.  Make  a  mental   picture  of  what  each  of  the  following 
expressions  represents,  and  tell  how  much  each  lacks  of  a  unit 
of  its  own  kind :  |-  of  a  pie ;  J  of  an  apple ;  T7^  of  a  foot ;  |-  of 
a  square  yard ;  f  of  a  regular  pentagon ;  £  of  a  regular  hexa- 
gon ;  J  of  a  2-inch  cube. 

6.  A  fraction  whose  numerator  is  less  than  its  denominator 
is  called  a  Proper  Fraction.     Give  some  proper  fractions  and 
tell  how  much  each  lacks  of  being  equal  to  the  whole  of  which 
it  is  a  part. 

7.  A  fraction  whose  numerator  is  equal  to  or  greater  than 
its  denominator  is  called  an  Improper  Fraction.      Give  some 
improper  fractions  and  tell  how  much  each  exceeds  one  unit. 

108 


FRACTIONS  109 

8.    Separate  the  following  fractions  into  two  lists,  one  of 
proper  fractions,  the  other  of  improper  fractions  : 

f  »    I      *     A     « 

H     tt       -3         «      V&      101% 

9.    Write  a  proper  fraction  whose  terms  are  5  and  7.    Write 
an  improper  fraction  with  the  same  numbers  as  terms. 

10.  A  number  that  consists  of  an  integer  and  a  fraction  is 
called  a  Mixed  Number  ;  as  3J.     How  many  halves  of  a  circle 
in  31  equal  circles  ?     Illustrate.     Does  the  following  explana- 
tion seem  to  you  to  be  true  ? 

As  there  are  2  halves  in  1  whole,  in  3  wholes  there  are  3  times  2  halves, 
or  6  halves,  6  halves  -f  1  half  =  7  halves. 

11.  Change  2|  to  an  improper  fraction   and   explain   the 
process. 

12.  Change  to  equivalent  improper  fractions  : 

7*        Si        8i      16|      Sf      7J      of      20J 
8*      15f      21f        6f      5*      1$      6J      llf 

13.  Tell  how  a  mixed  number  is  changed  into  an  equivalent 
improper  fraction. 

14.  Change  to  equivalent  improper  fractions  : 

7«      5lf      12f 
4f        6} 


15.  Write   a  mixed   number   whose   fractional    part   is   f. 
Change  it  to  an  equivalent  improper  fraction. 

16.  Write  a  mixed  number  whose  integral  part  is  7.    Change 
it  to  an  equivalent  improper  fraction. 

17.  CLASS  EXERCISE.  -  may  give  a  mixed  number,  and 
the  class  may  reduce  it  to  an  improper  fraction. 


110  FRACTIONS 

18.  How  many  wholes  are  there  in  f  ?  f  ? 

SUGGESTION  TO  TEACHER.  Lead  pupils  to  express  in  their  own  way 
the  evident  fact  that  since  it  takes  2  halves  to  make  a  whole,  there  will 
be  as  many  wholes  in  any  number  of  halves  as  there  are  groups  of  2  in 
that  number. 

19.  To   reduce   a  fraction  is  to  change  its  form  without 
changing  its  value.     Reduce  the  following  improper  fractions 
to  mixed  numbers  : 

f     ¥     -¥-     ¥     «     tt     «     W   W 

20.  Give  directions  for  reducing  Nan  improper  fraction  to  a 
whole  or  a  mixed  number. 

21.  Reduce  to  integral  or  mixed  numbers: 

m      Hi      W- 


22.  CLASS  EXERCISE.        —  may  give  an  improper  fraction, 
and  the  class  may  change  it  to  a  mixed  number  or  to  an  integer. 

23.  A  fraction  is  an  expression  of  division,     if-  equals  how 
many  units  ?     In  the  expression  -^  which  number  is  the  divi- 
dend ?    Which  is  the  divisor  ?     What  is  the  quotient  ?     Show 
the  same  with  regard  to  the  expression  -1-0-.    With  -^k    With  -:75-. 

SUGGESTION  TO  TEACHER.  Lead  pupils  to  see  that  proper  fractions 
also  express  division.  £  indicates  that  one  unit  is  divided  by  3.  Let 
lines  be  drawn  and  divided  to  illustrate  such  facts  as  that  ^  of  2  yd.  or  of 
6  ft.  equals  f  of  a  yd.  ,  that  f  of  a  ft.  or  9  in.  equals  %  of  3  ft. 

•\ 

24.  Draw  a  line  3  in.  long.     Divide  each  inch  into  fourths 
and  show  that  f  of  1  in.  equals  £  of  3  in. 

25.  When  the  numerator  and  denominator  of  a  fraction  are 
made  to  change  places,  the  process  is  called  inverting  the  frac- 
tion, as  f  inverted  is  f.    The  fraction  resulting  from  this  inver- 
sion of  a  fraction  is  called  the  Reciprocal  of  the  original  fraction. 
Thus  -f  is  the  reciprocal  of  f  .     What  is  the  reciprocal  of  the 
fractionf?    £?    T9T?    y~? 

26.  Which  is  greater,  f  or  its  reciprocal  ?    ^  or  its  reciprocal  ? 


FRACTIONS 


111 


FIG.  1. 


27.  The  ratio  of  Mr.  A's  money  to  Mr.  B's  money  is  -f  .  What 
is  the  ratio  of  Mr.  B's  money  to  Mr.  A's  money  ?  If  Mr.  A's 
money  is  $  8,  how  much  has  Mr.  B  ? 

28.  Draw  a  regular  octagon.  Divide  it 
by  diagonals  into  8  equal  isosceles  trian- 
gles. Divide  each  isosceles  triangle  into 
equal  right  triangles  as  in  Fig.  1.  Each 
isosceles  triangle  equals  what  part  of  the 
octagon  ?  Each  right  triangle  equals  what 
part  of  an  isosceles  triangle  ?  Of  the  octa- 
gon ?  i  of  i  =  ? 

4SuGGESTioN  TO  TEACHER.  Let  a  large  copy  of  Fig.  1  be  drawn  upon 
the  board  as  a  basis  for  the  following  exercises. 

9.    Find  from  the  figure  the  values  of  the  following: 
iofj  i'afj  i«fj  ioff 

£ofi  |off  |of|  JofJ 

30.  A  fraction  of  a  fraction  is  called  a  Compound  Fraction. 
What  is  the  value  of   the  compound  fraction  1  of  ^  when 
expressed  in  simple  form  ? 

31.  How  many  inches  equal  J  of  \  of  a  foot? 

32.  See  if  the  following  reasoning  is  true  : 

Since  of  => 


of      =  2  tiroes         or 

33.    The  following  rule  is  founded  upon  the  same  reasoning: 
To  find  the  simple  form  of  the  value  of  a  compound  fraction  — 
Find  the  product  of  the  numerators  for  the  numerator  of  the 

simple  fraction   and   the  product  of  the  denominators  for  its 

denominator.     Cancel  if  possible. 

By  the  same  reasoning  find  the  value  of  %  of  |. 


112  FRACTIONS 

34.  Simplify : 

a  b                          c                            d 

f  of  |  of  f  *  of  ^  of  f|       £  of  |  of  72       f  of  |  of  |  of  ^f 

e  f                            g                             h 

|of«  »of«                 A<*«             foffiof* 

<  j                             A;                         I 

f  off  of  if  foffofi^         fof235of2i     foffiofS 

m  no 

A  of  H  of  7J  fof^ofGi           |  of  |  of  8i 

35.  How  many  square  centimeters  in  ^  of  ^  of  a  square 
decimeter  ? 

36.  Mr.  King  owned  1  of  a  farm  and  sold  f  of  his  share. 
What  part  of  the  farm  did  he  sell  ?     If  there  were  200  acres 
in  the  farm,  how  many  acres  did  he  sell  and  how  many  had 
he  left  ? 

37.  Six  boys  divided  a  number  of  marbles  equally  among 
themselves.     Edward  Wells,  one  of  the  boys,  gave  ^  of  his 
share  to  his  younger  brother.     What  part  of  the  marbles  did 
Edward  keep  ?     If  there  were  54  marbles,  how  many  did  he 
keep? 

38.  Mr.   Hubbard  owned  |  of  a  mine  and  sold  f  of  his 
share.     If  the  mine  was  worth  $80,000,  how  much  did  he 
receive  ? 

39.  How  many  cubic  feet  in  f  of  f  of  a  cubic  yard  ?     In  f 
of  f  of  a  cubic  yard  ? 

40.  How  many  minutes  in  f  of  f  of  an  hour  ?     In  |  of  £  of 
an  hour  ? 

41.  -I  of  f  of  4  of  4  is  how  much  less  than  1  unit  ?     Than  2 

o          4          y  / 

units? 

42.  f  of  f  of  2f.  is  how  much  more  than  1  unit  ?     How  much 
less  than  3  units  ?     Than  8  units  ? 


FRACTIONS  113 

43.  Find,  from  the  octagon  on  page  111,  the  values  of  x  in 
the  following  equations : 

i=TZ6       t  =  f       i  =  f         i  =  TF       t  =  f         f  =  T<r 

fx  3  z  1   z  3  g  5  *  7  * 

=  Tff         T  —  ¥         ¥  —  T¥         "8  —  Tff         ¥  ~  T6"         t  ~  TF 

44.  In   changing   £   to   16ths,   by   what   number   are   the 
numerator  and   the  denominator  multiplied?     How  do  you 
find  it? 

45.  Change  i  to  a  fraction  of  equal  value,  having  a  larger 
denominator,   and    show  the   truth    of    the    following    prin- 
ciple : 

PRINCIPLE  1.  Multiplying  both  terms  of  a  fraction  by  the 
same  number  does  not  change  the  value  of  the  fraction. 

46.  Change  £  to  6ths.     f  to  6ths. 
r 

47.  Change  1,  f,  f,  and  £  to  12ths. 

48.  Change  f,  f,  ^  to  20ths.     J,  |,  f  to  18ths. 

49.  To  24ths  change  T%,  |,  |,  f,  f,  1. 

50.  Write  10  fractions  whose  value  is  \,  but  whose  denomi- 
nators are  all  different.     What  is  the  ratio  of  the  numerator 
of  each  fraction  to  its  denominator  ? 

SUGGESTION  TO  TEACHER.  Develop  the  idea  that  the  value  of  a  frac- 
tion does  not  depend  upon  the  magnitude  of  the  numbers  by  which  it  is 
expressed,  but  upon  their  ratio. 

51.  Write  7  fractions  whose  value  is  J,  and  whose  denomi- 
nators are  all  different.     What  is  the  ratio  of  the  numerator 
of  each  fraction  to  its  denominator  ? 

52.  Tell  how  you  change  fractions  to  equivalent  fractions 
expressed  in  higher  terms ;  that  is,  by  larger  numbers. 

53.  Change  \  to  hundredths  and  write  it  as  per  cent. 

54.  Change  to  hundredths  and  write  as  per  cent : 

********* 

HORN.    GRAM.    SCH.    AR. — 8 


114  FRACTIONS 

55.  When  T63-  is  expressed  as  J-,  what  has  been  done  to  each 
term  of  the  fraction  T6^  ? 

56.  Choose  some  fraction  whose  value  is  J,  although  ex- 
pressed  differently,    and   show   the   truth   of    the    following 
principle  : 

PRINCIPLE  2.     Dividing  both  terms  of  a  fraction  by  the  same 
number  does  not  change  the  value  of  the  fraction. 

57.  What  number  will  divide  both  terms  of  the  fraction  T6^  ? 
What  fraction  results  from  the  division  ? 

58.  In  this  way  reduce  to  lowest  terms  :    -J-J.     T57.     T9T.     -}-J. 

59.  Reduce  ||-  to  lower  terms,  but  not  to  its  lowest  terms. 

60.  What  is  meant  by  the  phrase,  "  reducing  a  fraction  to 
lowest  terms  "  ? 

61.  In  reducing  a  fraction  to  its  lowest  terms,  by  which  of 
the  common  divisors  of  both  terms  is  it  best  to  divide  them  ? 
Why? 

62.  Reduce  to  lowest  terms  and  tell  what  common  divisor 
you  use:    ft.     «.     ff     if     if     A-     if. 

63.  Reduce  to  lowest  terms: 

«  .«  «  H  «  H  « 
tt  If  A  H  i  A  H 
A  *f  H  ,  H  if  H  « 

64.  Tell  how  a  fraction  is  reduced  to  its  lowest  terms. 

65.  Find  by  continued  division  the  greatest  common  divisor 
of  147  and  196,  and  reduce  the  fraction  ||-J  to  lowest  terms. 


66.    Reduce  to  lowest  terms  : 

m      m      m      m 

T8TF  "252" 


FRACTIONS  115 

67.    How  many  inches  in  |f£  ft.  ?     In  f  f  ft.  ?     In  ff  J  ft.  ? 


68.  As  John  was  studying  fractions  one  evening,  his  uncle 
said  to  him,  "  If  you  will  go  on  an  errand  for  me  I  will  give 
you  f||f  of  a  dollar."     John  did  so,  and  received  the  money. 
How  much  was  it  ? 

69.  If  John's  uncle  had  given  him  fffffff  °^  a  ^0^ar?  now 
many  cents  would  he  have  received  ? 

70.  What  is  the  use  of  reducing  fractions  to  lower  terms  ? 

71.  CLASS  EXERCISE.     -  may  give  a  fraction  which  the 
class  may  reduce  to  lowest  terms. 

72.  Can  you  reduce  f  to  lower  terms  ?     Give  reason  for  your 
"Yes"  or  "No." 

73.  Write  a  fraction  whose  numerator  and  denominator  are 
both  prime  numbers.    Can  you  reduce  it  to  lower  terms  ?    Give 
reason  for  your  answer. 

74.  Numbers  that   have  no  common  factor  are  said  to  be 
prime  to  each  other,  although  they  may  be  composite  numbers. 
f  is  a  fraction  whose  terms  are  prime  to  each  other.     Are  the 
terms  prime  numbers  or  composite  numbers  ? 

75.  Write  5  different  fractions,  each  having  for  its  terms 
composite  numbers  which  are  prime  to  each  other. 

76.  Write  a  fraction  with  7  for  a  numerator  and  the  square 
of  7  for  the  denominator,  and  reduce  it  to  lowest  terms. 

77.  Write  a  fraction  with  the  square  of  8  for  the  numerator 
and  the  cube  of  8  for  the  denominator,  and  reduce  it  to  lowest 
terms. 

78.  A  fraction  whose  denominator  is  10,  or  any  other  power 
of  10,  may  be  expressed  as  a  decimal  or  as  a  common  fraction. 
Express  T\  as  a  decimal.     .50  as  a  common  fraction. 


116  FRACTIONS 

79.  Express  in  decimal  form  and  in  common  form  each  of 
the  following  :  Thirteen  hundredths  ;  one  hundred  sixty-seven 
thousandths;  two  thousand  six  hundred  seven  ten-thousandths; 
forty-three  thousandths;  six  hundred  fifty  -one  hundred-thou- 
sandths; forty  one  rnillionths. 

80.  Write  as  common  fractions  and  give  in  lowest  terms: 
.5.    .25.    .125.     99%.    75%.    .072.    .064.    5%.    40%.    80%. 
90%. 

81.  CLASS  EXERCISE.     -  may  give  a  decimal  fraction, 
and  the  class  may  express  it  as  a  common  fraction  in  its 
lowest  terms. 

ADDITION  AND   SUBTRACTION  OF  FRACTIONS 

82.  A  man  dying  left  3  sevenths  of  his  property  to  his  wife, 
3  sevenths  to  his  children,  and  the  remainder  to  a  library. 
What  share  of  it  did  the  library  receive  ? 

83.  1  ninth  -f-  4  ninths  +  2  ninths  =  ? 

84.  How  are  fractions  added  when  they  have  the  same  de- 
nominator ? 

85.  Fractions  that  have  the  same  denominator  are  called 
Similar  Fractions.     Write  three  similar  fractions  and  find  their 
sum. 

86.  Give  four  proper  fractions  that  express  13ths  and  find 
their  sum. 

87.  From  T6T  take  TV     TT-T3T  =  ? 

88.  Find  from  the  octagon  on  page  111  the  values  of  x  : 

1     i       3        _     se  1  3     __     a:  3  _     5     —  .  « 

Y  +  T3"  —  T5"  •        1  ~~  TS  —  TF  ?        IT  —  TS" 

¥  +  A  =  TS  t  ~~  T5B"  =  fV  f  ~~  A  =  TZ6 


90. 


ADDITION  AND   SUBTRACTION  OF  FRACTIONS       117 

91.  £  +  y7^  of  the  octagon  needs  how  much  to  complete  the 
figure  ? 

92.  In  adding  1  and  1,  how  must  i  be  changed?     Why  ? 

93.  By  what  number  must  each  term  of  the  fraction  \  be 
multiplied  to  change  the  fraction  to  12ths  ?     How  did  you  find 
that  number  ? 

94.  Can  the  terms  of  the  fraction  \  be  multiplied  by  an 
integer  that  will  change  it  to  llths? 

95.  Give  five  different  numbers  that  can  be  used  as  de- 
nominators for  fractions  whose  value  is  J. 

SUGGESTION  TO  TEACHER.  Lead  pupil  to  see  that  in  changing  a  frac- 
tion to  an  equivalent  fraction  having  a  larger  denominator  the  new 
denominator  must  be  a  multiple  of  the  old. 

96.  Change  %  to  15ths  by  the  following  rule : 

To  change  a  fraction  to  an  equivalent  fraction  having  a  larger 
denominator  — 

Multiply  both  terms  of  the  fraction  by  the  quotient  obtained  by 
dividing  the  new  denominator  by  the  old  denominator. 

97.  Change  J  to  an  equivalent  fraction  having  the  denomi- 
nator 21.     27.     24.     33.     36.      How  many   lOths  equal  f  ? 
How  many  30ths  ?     20ths  ?     35ths  ?     50ths? 

98.  Copy  the  following  equations,  substituting  for  x  its 
value : 

1  x  5  x  5       _    *  2  x  5  * 

T  =  T2"         T  —  T2"         2T  —  T2          TT  ~  T2"         6"  ~  VI 

99.  Find  the  least  number  that  is  a  multiple  of  4  and  7. 
Change  f  and  -f  to  equivalent  fractions  having  that  number  for 
their  denominators.     Add  those  fractious. 

100.  Change  |  and  f  to  equivalent  fractions  having  for  their 
denominators  the  least  common  multiple  of  8  and  5.  Add 
those  fractions. 


118  FRACTIONS 

101 .  Change  |-  and  ^  to  equivalent  fractions   having  the 
least  integral  denominator  that  they  can  both  use.     Add  the 
fractions. 

102.  Change  ^  and  -J  to  equivalent  fractions  having  for  their 
denominator  the  least  common  multiple  of  their  present  de- 
nominators.    Then  add  them. 

103o  The  least  common  denominator  that  two  or  more  frac- 
tions may  have  is  the  least  common  multiple  of  their  denomi- 
nators. Express  the  following  fractions  with  their  least 
common  denominator:  4.  f.  •£.  J^..  4-.  f- 

4  O  o  12  -  •> 

104.  Change  ^  and  ^  to  equivalent  fractions  having  their 
least   common   denominator,   and    explain    your    method    of 
changing  them. 

105.  Express   with   least    common    denominator    and   find 
values:   £+f.     f  +  £     f  +  A-     i  +  A- 

106.  Find  values : 

«  i-i  c  t-i  e  *-*  ?  t-t 

b   i-i-V  d   |-i  /|-i  h  }-| 

Add: 

a  6  c  d 

107.  i  i  i         i,  |,  }         i  i,  i         i  i,  f 

108.  i,  ^  |  i,  |,  -f-  ^  ^  *  i,  |,  f 

109.  1,  ^  i  i,  |,  f  ^  i,  TV  i,  f,  H 

no.  i,  *,  A  i  i  A  i  i-  i  i  i  f 

in.  f,  i  I  i  i,  A  f>  i  A  i  i  f 

112.  i,  |,  i  |,  $,  £  i  i,  i  i>  i  t 

113.  i,  |,  A  i  f  A  i  T>  H  i  i  A 

114.  f,  f,  A  i,  i,  i  I,  f,  f  f,  fc  A 

SUGGESTION  TO  TEACHER.  For  additional  practice  let  the  class  sub- 
tract the  second  fraction  from  the  sum  of  the  first  and  third  in  Examples 
107-114. 


ADDITION  AND   SUBTRACTION   OF   FRACTIONS        119 

116.  How  much  greater  than  ^  is  ^  ?    ^  ?    £  ?    £  ?    |? 

117.  &  is  how  much  more  than  -ft  ?    fa  ?    -fa  ?    fa  ?    fa? 

118.  CLASS  EXERCISE.        —  may  give  two  fractions,  and  the 
class  may  find  their  sum  or  their  difference  as  may  be  directed. 

119.  Draw  a  line  AB  and  mark  two  points  in  it,  C  and  D. 
If  AC  represents  \  a  mile,  CD  \  of  a  mile,  and  DB  -fj-  of  a 
mile,  what  distance  is  represented  by  AB  ? 

120.  John  spent  ^  of  his  money  for  a  top,  J  of  it  for  a 
ball,  and  £  of   it  for  candy.      What  part  of  it  had  he  left? 

121.  How  much  money  did  John  spend  for  each  article  and 
how  much  had  he  left  if  he  had  at  first  12^  ?    48^  ?     24^  ? 

122.  f  +  T\  is  how  much  more  than  f  —  T\  ? 

123.  The  sum  of  -I  and  4  is  how  much  more  than  their 

o  o 

difference  ? 

124.  Express  in  lowest  terms  the  ratio  of  15  to  20  and  of 
16  to  30.     Find  their  sum.     Their  difference. 

125.  Find  the  sum  of  -f-  and  its  reciprocal.     Of  f  and  its 
reciprocal. 

126.  Find  the  sum  of  |  and  its  reciprocal.     Find  the  differ- 
ence between  f  and  its  reciprocal. 

127.  What  is  the  square  of   10?     The  5th   power?      3d 
power  ? 

128.  Which  power  of  10  is  the  denominator  of  the  decimal 
.01?     .0015?     .003?     .00008?     .000009? 


129. 

Add  -j2^,  -^ 

jfa>,  TOTP      Add 

nr<p  &>  ii 

130. 

Add: 

a 

6 

c 

rf 

.07 

.165 

.06 

.485 

.018 

.2145 

.016 

.6 

.5 

.31 

.07 

.21 

120  FRACTIONS 

131.  When   several   fractions  have  denominators  that  are 
powers  of  10,  is  it  easier  to  add  them  as  common  fractions  or 
as  decimals  ?     Why  ? 

132.  Write  in  decimal  form  and  add : 

jflT  TflTFTF  nfinnF  T% 

133.  Write  in  decimal  form  and  find  values : 
TO"  ~~  T^TJ"     Tuinr  ~~~  TTIF     TTRT  ~o  ~~~  TTrluo" 

134.  Which  is  greater,  3  or  3.00  ?    7  or  7.000  ? 

135.  Change  f  to  a  decimal. 

3  reduced  to  lOOths  equals  3.00.  As  3  equals  3.00,  3  -r-  4  equals 
3.00  H-  4.  3.00  -r-  4  equals  .75.  Hence  f  =  .75. 

136.  Change  f  to  a  decimal  by  the  following  rule.    Explain. 
To  reduce  a  common  fraction  to  a  decimal  — 

Annex  ciphers  to  the  numerator  and  divide  by  the  denominator. 

137.  Change  to  decimals  and  add : 

t     i     *     A     «     it     i     «     A 

138.  Change  -^  to  a  decimal,  stopping  at  lOOOths. 

Common  fractions  cannot  be  reduced  to  exact  decimals,  if  when  re- 
duced to  their  lowest  terms  their  denominators  contain  any  factors  other 
than  2  and  5. 

139.  Reduce  the  following  fractions  to  decimals,  not  carry- 
ing the  work  beyond  ten-thousandths : 

i      r' .  ,  jt     i    -.H  '  « ..  ...t ..  if  •  -« 

m      H      If      A      A      A      A      A      W 

140.  Reduce  the  following  to  decimals  of  not  more  than  3 
places,  and  add  them.     If  the  division  is  not  exact,  make  the 
remainder  the  numerator  of  a  common  fraction : 

f*  TV  I  I  A  A  f  A 


ADDITION  AND   SUBTRACTION  OF  FRACTIONS        121 

141.  Tell  at  sight  which  of  the  following  fractions  can  be 
reduced  to  exact  decimals : 

A       $        A       I        TT        M        'fr        T£        M 

Eeduce  the  fractions  to  decimals  of  not  more  than  three 
places,  and  find  their  sum. 

142.  Divide  2  by  each  number  larger  than  itself  that  is 
expressed  by  one  digit.     Express  the  quotient  as  a  decimal  of 
not  more  than  three  places. 

143.  Divide  three  by  each  number  between  10  and  20,  and 
express  the  quotient  as  a  decimal  of  not  more  than  three  places. 

144.  Change  to  decimals,  stopping  at  hundredths : 

*    t    i    i    *    *    *    *    t     I 

145.  How  many  lOOths  or  per  cent  equal  $  ?  f?  |?  f  ?  |? 

146.  How  many  per  cent  equal  ^ ?     -f^ ?     ft?    |$  ? 

147.  How  many  per  cent  equal  ft  ?  ft?  ^?  ft?  ft  ?  |£  ? 

148.  What  per  cent  of  anything  is  \  of  it  ?     % ?     •fI?    -fa? 

149.  Add  2|  and  7$: 

7$       f  +  £  =  |  or  1|,  which  added  to  the  sum  of  2  and  7  =  10£. 
10J 

150.  Add: 

a  b  c  d  e 

7f  lOf  8|  6f  2| 

3f  6£  3^  8|  6ft 

8f  5^  4f  7j%  8J 

151.  Tell  how  the  following  mixed  numbers  are  added: 

abed 
8£  16f  66| 

18f  8J 

GJ:  12$  16| 


122  FRACTIONS 

152.  Add: 

91|        41f        93f        68f 

153.  CLASS  EXERCISE.     may  give  three  mixed  num- 
bers, and  the  class  may  find  their  sum. 

154.  A  farmer  used  77^-  acres  of  land  for  wheat,  40  J  acres 
for  corn,  If  acres  for  vegetables,  29^-  acres  for  pasturage,  and 

acres  for  an  orchard.     How  many  acres  were  in  the  farm  ? 


155.  Mr.  White  has  3  fences  on  his  farm;  one  is  168^-  rd. 
long,   another   456T4T  rd.   long,    and   another  328T5T  rd.   long. 
How  many  rods  of  fencing  has  he  in  all  ?     How  much  did  his 
fence  cost  at  75^  a  rod  ? 

156.  After  selling  3f  acres,  a  farmer  had  left  123 j  acres. 
How  much  land  had  he  at  first  ? 

157.  How  long  is  it  from  half  past  eight  A.M.  to   noon? 
From  quarter  before  nine  to  half  past  eleven  ?     From  quarter 
past  two  to  quarter  to  six?     From  half  past  ten  to  quarter 
past  one  ?     From  a  quarter  of  eleven  to  half  past  three  ? 

158.  From  14  We  subtract  \  from  1  of  the  units  of  the  minu- 
take       7-i-     en(^'     '^^ie  rema'nder  is  \.    As  1  unit  has  been 

^     subtracted  from  the  4  units  3  units  are  left.    7  from 

6£     13  leaves  6. 

a  b  c  d  e  f 

159.  From     8  9  18  25  16  20 
take       6£          5|            4f            3|          11£  4£ 

160.  Mary's  aunt  sent  her  6  yd.  of  cashmere  for  a  dress. 
5|  yd.  were  used.     How  many  yards  were  left  ? 

161.  In  a  jumping  match  Thomas   jumped  3  ft.  and  his 
brother  jumped  2-jAj-  ft.     How  much  farther  did  Thomas  jump 
than  his  brother  ? 

162.  Make  problems  in  which  a  mixed  number  is  subtracted 
from  an  integer. 


ADDITION  AND   SUBTRACTION  OF   FRACTIONS        123 

163.    Subtract  : 

a  b  c  d  e  f  g 


2$         3&         2|          2j_         3|          2|          3i 

164.  A  piece  of  string  3^-  ft.  long  was  cut  from  a  piece 

ft.  long.     How  much  was  left  ?      5^  ft.  were  cut  from  the  re- 
mainder.    How  much  then  remained  ? 

165.  From  41  take  -f. 

4i 
2  As  |  cannot  be  subtracted  from  |,  we  subtract  f  from  1|. 

_Z       This  leaves  only  3  units  in  the  minuend. 
3f 

166.  Find  difference  : 

a  b  c  d  e  f  g 

8|          6i          8J          6i          8f          6$  7i 

1         J         J          J          J         J  J 

167.  Find  difference  : 

a  b  c  d  e  f  g 


h  i  j  k  I  m 

S$         9i          7f          8i          4|          6i 
3|          2|         4f          2|          2|          2| 

168.  Write  two  mixed  numbers  whose  fractional  parts  have 
the  same  denominator.     Let  the  mixed  number  whose  integral 
part  is  the  larger  have  the  smaller  fractional  part.     Find  dif- 
ference between  the  mixed  numbers. 

169.  Find  difference: 

a  b  c  d  e  f  g 

3J  or  -|       7J          8^ 
2f 


124  FRACTIONS 

170.  Write  two  mixed  numbers  the  fractional  parts  of  which 
have  different  denominators.     Let  the  mixed  number  that  has 
the  larger  integral  part  have  also  the  larger  fractional  part. 
Find  their  difference. 

171.  CLASS  EXERCISE.     may  give  two  mixed  numbers 

like  those  described  in  Ex.  170,  and  the  class  may  find  their 
difference. 

172.  Find  difference : 

a  b  c  d  e  f  g 

91  97 

"5"  9 


I  m  n 

6£  7£  4| 

2J  2f  21  4^  4T\          C2\  2* 

173.  Write  two  mixed  numbers  whose  fractional  parts  have 
different  denominators.     Let  the  mixed  number  whose  integral 
part  is  the  greater  have  the  smaller  fractional   part.     Find 
their  difference. 

174.  If   a  line  is  3J  in.  long,  how  many  inches  must  be 
added  to  make  it  5 J  in.  long  ?     Represent. 


175.  3^ 

176.  Find  values  of  x: 

a    10£-7i  =  x.  e     9    -3%  =  x. 

b      6    -  4£  =  x.  f  10£  -  5|  -  x. 

c      7    -3%  =  x.  g     7$-2±=x. 

4    =  z.  h   12    -7=a. 


177.  A  weighs  148  J  lb.,  B  157f  lb.,  C  1611  lb.,  D  175|  Ib. 
How  much  do  they  all  weigh  ? 

178.  What  is  the  difference  between  the  weights  of  A  and 
C?     AandB?     AandD?     B  and  D  ?     B  and  C  ?     CandD? 


ADDITION  AND   SUBTRACTION   OF  FRACTIONS       125 

179.  Mr.  Otis  rode  23f  miles  on  Monday,  llf  miles  on  Tues- 
day.   On  Wednesday  he  rode  as  far  as  on  Monday  and  Tuesday. 
How  far  did  he  ride  in  the  three  days  ? 

180.  Mr.  Carr  rode  on  his  bicycle  to  a  city  91f  miles  dis- 
tant.    The  first  day  he  rode  16£  miles ;  the  next  day  he  rode 
3^  miles  more  than  on  the  first  day.     On  the  third  day  he  rode 
21  miles  more  than  on  the  second  day.     How  far  did  he  ride  in 
those  three  days  ?     How  many  miles  more  did  he  ride  before 
he  reached  the  city  ? 

181.  Mr.  Grey  planted  75J  acres   in  wheat,  45f  acres   in 
corn,  and  7-J  acres  in  oats.     How  many  acres  of  grain  did  he 
cultivate  ? 

182.  In   a   township  containing  23,039T^  acres,  the   roads 
occupy  345f  acres,  and  the  rest  is  divided  into  farms.     How 
many  acres  in  the  farms  of  that  township? 

183.  A  stove  burned  180f  Ib.  of  coal  in  one  week,  175 J  Ib. 
in  another  week,  and  205^  Ib.  in  another  week.     How  many 
Ib.  did  it  burn  in  the  three  weeks  ? 

184.  A,  B,  and  C  own  a  mine.     A  owns  T5^  of  it,  B  owns  f  of 
it,  and  C  owns  the  rest.     How  much  does  C  own  ? 

185.  If  the  mine  is  worth  f  248,400,  what  is  the  value  of 
each  man's  share  ? 

186.  A  has  75f  acres  of  land,  B  has  13|  more  acres  than 
A  and  4|  acres  less  than  C.     How  many  acres  has  B  ?     C  ?     A 
and  B  ?     A  and  C  ?     B  and  C  ? 

187.  A  farmer  has  a  field  in  the  form  of  a  trapezoid.     One 
of  the  parallel  sides  is  71T4T  rd.  long,  the  other  is  68|  rd.  long. 
Of  the  non-parallel  sides,  one  is  53f  rd.  and  the  other  is  54^-  rd. 
Represent  and  find  length  of  perimeter  of  the  trapezoid. 


126  FRACTIONS 

MULTIPLICATION  OF  FRACTIONS 

188.  Multiply  £  by  £. 

To  multiply  any  number  by  ^  is  to  take  £  of  it. 

189.  By  1  multiply : 

f         f         !         TGT         I         I         f 

190.  Make  a  rule  for  multiplying  a  fraction  by  a  fraction. 

Multiply  : 

abed 

1  Q 1  4    \/    7  .2.1   V     8  1  5    y    49  6     y    5  g 

1  qo  4  5    v    1  §  5  0    v    1  1  1  8    v    7  27    v    8 

IVA.  -Q-f   A   YT  77   A   TJ-J  T5T   A  IF  32    *   "9 

194.        14  x  44         M  x  -rV         M  x  -U         *«-  x  M 


195.  CLASS  EXERCISE.     may  give  two  fractions  and 

the  class  may  find  their  product. 

196.  CLASS  EXERCISE.     may  give   a   proper   fraction 

and  an  improper  fraction  and  the  class  may  find  their  product. 

197.  CLASS  EXERCISE.     may  give  three  fractions  of 

such  a  kind  that  cancellation  may  be  used  in  finding  their 
product  and  the  class  may  find  the  product,  canceling  wherever 
possible. 

198.  Multiply  |  by  itself. 

199.  Square : 

2  4    5    5    10     12     3     151917    25    41 

3  o     7     9     11     la      7      16     20     18     30     53 

200.  What  part  of  a  square  inch  is  a  rectangle  that  is  J- 
of  an  inch  long  and  ^  of  an  inch  wide  ?     Draw  a  figure  and 
prove  your  work. 

201.  What  part  of  a  square  yard  is  a  rectangle  ^  of  a  yard 
square  ?     How  many  square  feet  in  it  ? 

202.  Multiply  f  by  the  square  of  f . 


MULTIPLICATION  OF  FRACTIONS  127 

203.  Cube: 

t        I        *        *        I        I        I         A         T4T        A 

204.  Draw  a  rectangle  whose  length  is  £  of  a  foot  and  width 
i  of  a  foot.    What  fraction  of  a  square  foot  is  its  area  ?    Prove 
by  reducing  the  fractions  of  a  foot  to  inches. 

205.  What  fraction  of  a  square  foot  is  a  rectangle  f  of  a  foot 
long  and  ^  of  a  foot  wide  ?     How  many  square  inches  in  it  ? 

206.  How  long  is  the  perimeter  of  a  rectangle  £  of  a  foot 
long  and  ^  of  a  foot  wide  ?     Give  the  area  of  the  rectangle 
in  fractions  of  a  square  foot,  and  also  in  square  inches. 

207.  How  many  square  feet  in  a  square  f  of  a  foot  in 
dimensions  ?     How  many  square  inches  ? 

208.  How  many  square  feet  in  a  square  f  of  a  foot  in  dimen- 
sions ?  How  many  square  inches? 

209.  Add  the  product  of  ^  x  f  to  the  product  of  f  x  7. 

210.  Subtract  the  product  of   f  x  £  from  the  product  of 

I  x  I- 

211.  The  product  of  several  numbers  is  called  their  con- 
tinued product.    What  is  the  continued  product  of  3,  5,  and  7  ? 
Of  i  |,  and  f?    Of  £,  24T,  and  f  ?     Offhand}? 

212.  Multiply  |£  xf 

Observe  that  the  denominator  of  the  fraction  f  may  be  omitted  without 
changing  the  result. 

Multiply : 

a  b  c  d  e 

213.  f  x  24         ^  x  35         5  x  27         J  x  64       T2T  x  33 

214.  ^  x  100      f  x  18        f  x  21      ^  x  18        |  x  24 

215.  36  x  fV       48  X  I        32  x  A      39  x  A       60  x  ii 

216.  77  x  T4T      42  x  f        63  x  2^      54  x  ^      65  x  fV 

217.  Multiplying  the  numerator  of  a  fraction  by  an  integer 
has  what  effect  upon  the  value  of  the  fraction  ?     Illustrate. 


128  TRACTIONS 

218.  Dividing  the  denominator  of  a  fraction  by  an  integer 
has  what  effect  upon  the  value  of  the  fraction  ?     Illustrate. 

219.  Give  either  of  two  ways  by  which  a  fraction  may  be 
multiplied  by  an  integer. 

220.  Multiply  12  by  a  proper  fraction.     Is  the  product 
greater  or  less  than  12  ? 

221.  Multiply  12  by  an  improper  fraction.     Is  the  product 
greater  or  less  than  12  ? 

222.  When  will  the  product  of  an  integer  and  a  fraction 
be  greater  than  the  integer  ? 

223.  Multiply  64  by  3J. 

64 
8* 

192  =  the  product  of  64  and  3. 
16  =  the  product  of  64  and  £. 

208  =  the  product  of  64  and  3J. 


Multiply  : 

a 

b 

c 

d 

224. 

55 

164 

164 

125 

4i 

21 

5t 

5 

225.          72  81  208  144 


226.          81  115  64  343 

H 


227.  66                  201  192  512 

51                  11}  4f  7| 

228.  172                   981  111  169 

21  101 


MULTIPLICATION   OF  FRACTIONS  129 

a  b  c  d 

229.       385  243  78  408 


230.  Multiply  an  integer  by  a  mixed  number  and  explain 
your  method. 

231.  At  60^  a  yard,  what  is  the  cost  of  7£  yd.  of  silk  ?    8£ 
yd.?     lOfyd.?      12f  yd.  ?      14|  yd.  ?      15|  yd.  ?     18f  yd.  ? 

232.  160  square  rods  equal  1  acre.     How  many  square  rods 
in  |  of  an  acre?     In  |  A.  ?     ^  A.  ?     |f  A.  ?     ^  A.  ?     £  A.  ? 

233.  Mr.  Hill  has  a  lot  40  rd.  long  and  3  rd.  wide.     What 
fractional  part  of  an  acre  is  it  ? 

234.  How  many  square  rods  in  2|  A.  ?     5|  A.  ?     7T%  A.  ? 

A.? 

235.  Multiply  16 $  by  6. 


Multiply  : 
a 

16f 
6 

96  =  the  product  of  16  and  6 
4  =  the  product  of  f  and  6 

d 

100  =  the  product  of  16f  and  6 
b                       c 

236.       15J 
8 

92J                  39| 
24                    12 

122| 
22 

237.     124f 
16 

72|                164| 
15                      4 

109J 

27 

238.       24| 
9 

119f                  81$ 
15                    16 

98| 
5 

HORN.    GRAM.    8CH.    AR. 9 


130  FRACTIONS 

239.  Tell  how  a  mixed  number  is  multiplied  by  an  integer. 

240.  At  18f  ^  per  yard,  what  is  the  cost  of  4  yd.  of  gingham  ? 
7yd.?     8yd.?     10yd.?     12yd.? 

241.  How  long  is  the  perimeter  of  an  equilateral  triangle,  a 
side  of  which  is  4T72  ft.  long  ? 

242.  If  each  side  is  4T9^  ft.  long,  how  long  is  the  perimeter 
of  a  rhombus  ?     Of  a  regular  octagon  ?     Of  a  regular  hexa- 
gon ?     Of  a  regular  pentagon  ? 

243.  At  331  ^  per  yard,  what  is  the  cost  of  8  yd.  of  dress 
goods  ?     9  yd.  ?     12  yd.  ?     14  yd.  ?     15  yd.  ?     18  yd.  ? 

244.  Multiply  2^  by  3^. 

Before  small  mixed  numbers  are  multiplied  together,  they  should  be 
reduced  to  improper  fractions. 

fx¥  =  Y  =  8i 

Multiply : 

245.  a  21  by  3J  d     5|  by  12J 
b   3J  by  6J  e  18f  by  11| 
c  4f  by  6^                          /    7i  by  66| 

246.  Square: 

2|       H       3i       2}       6i       If       If       2f 

47.    Cube: 

11      H      34      31      4i      6J      3f      2^ 

248.  CLASS   EXEKCISE.      may  give  two  small  mixed 

numbers,  and  the  class  may  find  their  product. 

249.  CLASS  EXERCISE.     may  give  a  small  mixed  num- 
ber, and  the  class  may  find  its  square. 

250.  Draw  on  the  floor  a  square  5£  yd.  in  dimensions.   Each 
side  is  1  rd.  long.     Find  the  number  of  square  yards  in  a  square 
rod. 


MULTIPLICATION   OF   FRACTIONS  131 

251.  How  many  feet  long  is  a  rod?     Find  the  number  of 
square  feet  in  a  square  rod. 

252.  Each  of  the  short  sides  of  a  rectangle  is  7-f  in.      The 
long  sides  are  9|  in.  each.    Find  the  area  of  the  rectangle. 

253.  One   side  of  a  rectangle  is  llf  in.,  and  each  of  its 
adjacent    sides   is   3£  in.    shorter.      Find    the   area   of   the 
rectangle. 

254.  Draw  on  paper  or  pasteboard  a  circle  whose  radius  is 
3£  in.     Cut  it  out.     By  measuring  the  circumference  with  a 
tape  measure,  it  will  be  found  to  be  nearly  22  in.      %f-  is 
considered  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter.     What  is  the  ratio   of  a  diameter  to  the  circum- 
ference ? 

255.  Find  the  circumference  of  a  circle  whose  diameter  is  7 
in.     21  in.     14  in. 

256.  How  long  is  the  circumference  of  a  circle  whose  radius 
is  21  in.  ?     3£  in.  ?     10£  in.  ?     17J-  in.  ? 

257.  How  long  is  the  circumference  of  the  largest  circle  that 
can  be  cut  from  a  piece  of  paper  4-|  in.  square  ? 

258.  A  round  flower  bed  14  ft.  across  has  a  border  of  pinks, 
set   6   in.    apart.      How  many   pinks   in  y^  of   the  border  ? 
Represent. 

259.  Mrs.  Smith's  wash  bench  is  4  ft.  long  and  If  ft.  wide. 
A  tub  is  set  upon  it  in  such  a  way  that  the  lowest  hoop  of  the 
tub  touches  the  front  edge  and  also  the  back  edge  of  the  bench 
without  extending  over  either  edge.    What  is  the  circumference 
of  the  hoop  ? 

260.  A  round  tin  pail  with  straight  sides  is  8  in.  across  and 
10  in.  high.     How  long  is  the  diameter  of  the  largest  plate  that 
can  be  placed  on  the  bottom  of  it  ?     The  circumference  ? 

261.  If  a  ball  is  cut  into  two  equal  parts  by  one  cut,  what  is 
the  shape  of  the  flat  surface  of  each  part  ? 


132 


FRACTIONS 


262.  A  round  apple  4  in.  in  diameter  was  cut  into  halves. 
One  of  the  halves  was  laid  with  its  flat  side  down  upon  a 
plate,  in  such  a  way  that  no  part  of  the  cut  surface  of  the 
apple  extended  beyond  the  plate.     Find  the  diameter  and  the 
circumference  of  the  smallest  plate  that  could  be  used  for  that 
purpose. 

263.  A  part  of  a  circumference  is  called  an  Arc.     Draw  a 
circle  and  divide  its  circumference  into  several  arcs. 

264.  If  the  radius  OA  (Fig.  2)  is  3  in., 
how  long  is  the  diameter  AC?    How  long 
is  the  circumference  ?    . 

265.  What  part  of  the  circumference  is 
the  arc  AC?     How  long  is  it  ? 

266.  The  arc  AB  is  %  of  the  circum- 
ference.    How  long  is  it? 

FIG.  2.  267.    What  part  of  the  circumference  is 

the  arc  BC  ?     How  long  is  it  ? 

268.  If  the  radius  of  a  circle  is  5  in.,  how  long  is  the 
diameter?  The  circumference?  An  arc  which  is  JT  of  the 
circumference  ?  of  the  circumference  ? 


269.   Find  the  length  of  an  arc  which  is  -^ 
ference  of  a  circle  whose  radius  is  5  in.     12£  in. 


of  the  circum- 
37£ in. 

270.  The    circumference    ABCD    is 
divided    into    how   many    equal    arcs? 
If  the  diameter  of  the  circle  ABCD  is 
7  in.,  how  long  is  the   arc  AB?     The 
arc  ABC?     The  arc  BCD?    The  arc 
ABCD? 

271.  An  arc  which  is  £  of  a  circum- 
ference   is    called   a  Quadrant.     If    the 
diameter    of   the    circle    ABCD    were 

28  cm.,  how  long  would  a  quadrant  be? 


MULTIPLICATION  OF  FRACTIONS  133 


272.    Multiply  124f  by  6J. 
124f 


744  =  the  product  of  124  by  6. 

62  =  the  product  of  124  by  J. 

4  =  the  product  of  f  by  6. 

£  =  the  product  of  f  by  £. 

810£  =  the  product  of  124|  by  6£. 

273.  This  method  of  multiplying  mixed  numbers  together  is 
useful  when  the  numbers  are  large.     Can  you  see  why  ? 

Multiply : 

a  b  c  d 

274.  441£  344£f  288f  456f 

16f 


abed 
275.    819f  64|  816^ 

9f  28^ 


276.  Square  64f.     32|.     24f     22T2T.     36J. 

277.  Cube  12J.     16J.     16J.     14|. 

278.  At  the  rate  of  17f  mi.  per  hour,  how  far  will  a  steam- 
boat go  between  nine  o'clock  Monday  morning  and  half  past 
ten  on  Tuesday  morning  ? 

Find  the  cost  : 

279.  Of  51  yd.  of  cloth  at  $  4|  a  yard. 

280.  Of  7-J-  A.  of  land  at  $  24^  an  acre. 

281.  Of  7|  T.  of  hay  at  $  21  J  a  ton. 

282.  Of  10J  yd.  of  ribbon  at  $  .23  a  yard. 

283.  Multiply  ^  by  fa  and  write  the  product  in  decimal 
form. 


284.   Find  the  product  of  -}-£  and  -f^  and  express  it  as  a 
mixed  number. 


134  FRACTIONS 

285.  Express  fj  and  T8^  in  decimal  form  and  find  their 
product. 

286.  If  asked  for  the  product  of  two  fractions  having  for 
denominators  some  power  of  10,  would  you  find  it  easier  to 
multiply  them  as  common  fractions  or  as  decimals  ?     Why  ? 

287.  Give  a  rule  for  pointing  off  the  product  of  two  decimals. 

288.  Let  a  =  .04,  b  =  .02,  c  =  .005,   d  =  .0007,   e  =  .00002, 
/=  .3.     Find  the  value  of  ab,  ac,  ad,  ae,  af,  be,  bd,  be,  bf,  cd,  cf. 

DIVISION  OF   FRACTIONS 

289.  How  many  times  is  1  fourth  of  an  inch  contained  in  3 
fourths  of  an  inch  ?    1  of  anything  in  £  of  it  ?  |-^-i=?  |-=-i=? 

290.  6  sevenths  -v-  2  sevenths  =?     f  -j-  -f  =  ?     10  elevenths 
^-  5  elevenths  =  ?    |f  -*-  ^  =  ? 

291.  A-nA=? 

T8T  contain  ^  as  many  times  as  8  units  of  any  kind  contains  3  units  of 
the  same  kind.     8  -j-  3  =  f  or  2|. 

292.  What  are  similar  fractions  ?     Illustrate. 

293.  Divide  |f  by  a  similar  fraction. 

294.  CLASS  EXERCISE.    may  give  two  similar  fractions, 

and  the  class  may  divide  the  greater  by  the  less. 

295.  How  is  a  fraction  divided  by  a  similar  fraction  ? 

296.  Use  |-  as  a  dividend  and  %  as  a  divisor. 

297.  Multiply  |  by  its  reciprocal. 

298.  Multiply  several  fractions   by   their   reciprocals   and 
compare  the  results. 

299.  How  many  times  is  f  of  anything  contained  in  f  of 
the  same  thing ?     £inj?     |in|? 

300.  Which  is  greater,  } -j-  f  or  j  x  | ?     }-5-'{orJx^? 


DIVISION  OF  FRACTIONS  135 

301.  $dividedbyf=?    f  multiplied  by  the  reciprocal  of  f=? 

302.  Think  of  two  similar  fractions.     Divide  the  larger  frac- 
tion by   the   smaller.      Compare   the  result  with  the   result 
obtained  by  multiplying  the  larger  fraction  by  the  reciprocal 
of  the  smaller.     Think  of  two  fractions  that  are  not  similar. 
Reduce  them  to  similar  fractions  and  make  the  same  com- 
parison.    Continue  this  until  you  see  the  reason  for  the  fol- 
lowing rule : 

To  divide  a  fraction  by  a  fraction  — 

Multiply  the  dividend  by  the  reciprocal  of  the  divisor. 

303.  By  this  method  divide  -^  by  ^. 


Divide 

:         a 

b 

c 

d 

304. 

A  by  f 

&  by  \ 

&byf 

|by  | 

305. 

A  by  T 

«by* 

if  by  A 

iby  | 

306. 

A  by  | 

f  by  f 

ifby  | 

Hby  - 

307. 

ft  by  i7 

ft  b>    A 

ffby  | 

Mby 

308. 

«  by  ff 

M  by  f 

1  4    VvV      7 
"2TT      J    10" 

Hby 

309. 

1  by  TT- 

Mby.fr 

M  by  A 

if  by 

310 

Wby  | 

tf  by  A 

ff  by  || 

liby  - 

311.  How  do  you  find  the  ratio  of  one  number  to  another  ? 
Find  the  ratio  of  f  f  to  ^-.     To  ^-.     To  -fa. 

Find  ratios : 

abed 

312.  A:f  tt'f  H:li  tt:H 

313.    &•*          H:A         A:f          H^ 

314.  |f:A  If^A  *:*  if  •  I 

315.  Multiply  the  numerator  of  a  fraction  by  an  integer. 
Is  the  value  of  the  fraction  increased  or  decreased? 


136  FRACTIONS 

316.  Find  by  trial  how  the  value  of  a  fraction  is  changed  by 
multiplying  its  denominator.     By  dividing  its  numerator.     By 
dividing  its  denominator. 

317.  Which  gives  the  greater  quotient,  16  divided  by  8,  or 
16  multiplied  by  |  ? 

318.  |f -5-8  =  ? 

We  may  write  8  as  f .     The  question  is  then  £f  -4- f  =  what  ?    This  is 
solved  by  the  rule  for  dividing  one  fraction  by  another. 

Find  values  of  x : 

a  b  c 

319.  f-=-2  =  a;        |T  -j-  3  =  a;          T67  -7- 12  =  a 

321.  f  -r-  4  =  a;        y2¥^-8  =  aj          fj  -j- 11  =  a 

322.  Divide  28  by  -f. 
Consider  28  as  *£. 

323.  Divide  24  by:      f  f  T«T.     f.  f      |f       f 

324.  Divide  41  by :      f .  f .  f      f .  f .      ||.      |f 

Divide  after  reducing  mixed  numbers  to  improper  fractions : 
a  b  c  d 

325.  3|-T-2i-          7J-*-18f        8J-f-lf  5f-r- 

326.  2^-f-3J         13i-f-4|          6|- 

327.  3£-i-6J         12£  ^-5f        37|- 

328.  CLASS  EXERCISE.      may   give    a    small    mixed 

number  for  a  dividend  and  another  small  mixed  number  for 
a  divisor,  of  such  a  kind  that  cancellation  may  be  used  in 
finding  the  quotient.     The  class  may  find  their  quotient. 

329.  At  a  picnic  5^-  pies  were  divided  equally  among  44 
persons.     What  part  of  a  pie  did  each  receive  ? 


DIVISION   OF   FRACTIONS  137 

330.  Mr.  Tod  has   13^   acres    devoted  to  celery,  which  is 
just  four  times  as  much  as  his  brother  has.     How  many  acres 
of  celery  has  his  brother  ? 

331.  The  top  of  a  newel  post  is  an  octagon  whose  perimeter 
is  l|i  ft.     How  long  is  one  side  of  the  octagon  ? 

332.  The  circumference  of  a  circle  is  9f  in.     How  long  is  its 
diameter  ?     Its  radius  ?     Eepresent. 

333.  Find  the  length  of  the  diameter  of  a  circle,  the  circum- 
ference of  which  is  11    in. 


334.  A  quadrant  of  a  circle  is  4^  in.     How  long  is  the  cir- 
cumference ?    Diameter  ?    Eadius  ? 

335.  For  75^,  how  many  yards  of  lace'  can  be  bought  at  21  f 
per  yard?   At  3^?   At  6^?   At8i^?  At  12^?  At  16.}?? 
At  37J^?     At  621^?     At66|^?     At  83^?    At 


336.  If  a  coat  costs  3  j  dollars,  how  many  coats  may  be  bought 
for  62£  dollars  ? 

337.  If  1J  yd.  of  cloth  are  required  for  a  coat,  how  many 
coats  can  be  made  from  87-J-  yd.  ? 

338.  If  Jerry  walks  2^  mi.  an  hour,  in  how  many  hours  will 
he  walk  7-J-  mi.  ?    11  mi.  ?    1  mi.  ?    |  mi.  ?    J  mi.  ?    1  mi.  ? 

339.  What  number  divided  by  3  will  give  5  for  a  quotient  ? 
What  fraction  divided  by  3  will  give  f  for  a  quotient  ?    Prove. 

340.  What  mixed  number  multiplied  by  3  will  give 

181?    6f? 


341.   A  fraction  that  has  a  fraction  in  one  or  both  of  its  terms 

1    2~    7 
is  called  a  Complex  Fraction  ;  as,  |,  =f>  ^T.     Write  a  complex 

fraction  whose  numerator  is  a  mixed  number,  and  denominator 
a  whole  number. 

|  is  read  "|  divided  by  iv 


138  FRACTIONS 

342.  Write  and  read  complex  fractions  as  follows : 

a  Numerator  an  integer,  denominator  a  simple  fraction. 

b  Numerator  a  simple  fraction,  denominator  a  mixed  num- 
ber. 

c    Numerator  an  integer,  denominator  a  mixed  number. 

d  Numerator  and  denominator  both  simple  fractions.  Both 
mixed  numbers. 

343.  A  complex  fraction,  like  other  fractions,  is  merely  an 
expression  of  the  division  of  the  numerator  by  the  denominator, 
and  it  is  reduced  to  a  simple  fraction  by  performing  that 
division;  as, 

f=H°rIxH- 

3 

Reduce  to  their  simplest  form  the  following  complex  fractions: 
a        b        c        d        e       f 

4}     U     3£     1}     3f     17 

3J      4|       T      5|      13      8j 

344.  Simplify: 

a          b          c          d          e        f       g        h 
^12J16J41|2|-3J6J7J 
33!      83j      87j      91|      20      30      50      60 

345.  CLASS  EXERCISE.     may  give  the  hardest  complex 

fraction  that  he  wrote  in  Ex.  342,  and  the  class  may  reduce 
it  to  a  simple  fraction. 

346.  The  product  of  two  numbers  is  15.     One  of  them  is  3. 
What  is  the  other  ?     How  is  it  found  ? 

347.  The  product  of  two  fractions  is  -fa.     One  of  them  is 
f .     What  is  the  other  ? 

348.  CLASS  EXERCISE.     may  give  the  product  of  two 

fractions  and  one  of  the  fractions.     The  class  may  find  the 
other  fraction. 


DIVISION   OF  FRACTIONS  139 

349.  Divide: 

liv  ~^  iV         T7TO"0~  ~*~  TO"         10000  "*"  TOO"         1000  ~*~  TOTF 

350.  Reduce  the  same  fractions  to  decimal  form  and  find 
the  quotients.     Which  is  the  easier  way  of  dividing  in  this 
case  ?     Why  ? 

351.  Leta=.4,  6  =  .08,  c=.032,  d=.0016.     Find  values  of : 

a      a      a      b       b      6 
b      c       d      a      c       d 

£.      -      £      f?      -      - 
a      b       d      a      b      c 

352.  Change  f  and  ^f  to  decimals  and  divide  the  greater  by 
the  less. 

353.  Change  to  decimals  and  divide :    f  by  J.     f  by  |. 

354.  If  4.5  yd.  of  silk  cost  $  6.75,  how  much  will  1  yd.  cost  ? 
3.7  yd.  ?     6.75  yd.  ? 

355.  Mr.  K  bought  a  lot  in  Washington,  D.C.,  for  $4500, 
paying  $1.875  per  square  foot.     How  many  square  feet  in  the 
lot? 

356.  If  1.7  yd.  of  cloth  is  used  to  make  -a  coat,  how  many 
coats  can  be  made  from  81.6  yd.  ? 

357.  How  much  cloth  at  $  .75  a  yard  can  be  bought  for 
$  45.75  ? 

358.  At  the  rate  of  8.25  mi.  an  hour,  in  how  many  hours 
will  a  stage  run  125  mi.  ? 

359.  If  a  barrel  of  beef  costs  $  14.25,  how  many  barrels  can 
be  bought  for  $  798  ? 

360.  Traveling  215.6  mi.  a  day,  in  how  many  days  will  a 
steamer  go  1000  mi.  ? 

361.  If  a  dollar  gains  5  i  interest  each  year,  in  how  many 
years  will  it  gain  another  dollar  ? 


140  FRACTIONS 

362.    In  how  many  years  will  $1.00  double  itself  at  4%  ? 


363.  One  third  of  John's  money  is  if.     How  much  has  he  ? 

364.  Thomas  spent  f  of  his  money  and  had  5  f  left.     How 
much  had  he  at  first  ?     Explain. 

365.  How  much  money  has  a  boy  who  can  spend  J  of  his 
money  and  have  left  70?    90?    20?    120?    $1.00?    $8.00? 

366.  How  much  money  has  a  boy  who  after  spending  30 
will  have  left  f  of  his  money  ?     J  ?     T97  ?     £?     f  ?     }  ? 

367.  Three  fourths  of  John's  money  is  150.     How  much  is 
£  of  it  ?    How  much  is  the  whole  ? 

368.  6=  f  of  what  number  ? 

SOLUTION  BY  ANALYSIS.     As  3  fifths  of  the  number  =    6 

1  fifth   of  the  number  =    2 
5  fifths  of  the  number  =  10 

369.  8  is  f  of  what  number  ?     Analyze  as  above. 

370.  Analyze.     12  is  f  of  what  number  ?    4.  of  what  ? 

371.  Find  values  of  x.     Analyze. 

a  -ff  of  x  =  14  e   f  of  x  =  15  i     f  of  x  =  18 

b     |  of  a?  =  10  /  |  of  x  =  16  J  V  of  x  =  20 

c     f  of  a  =  25  g   f  ofx  =  28  fc    J  of  a;  =  28 

d     f  of  a;  =  24  h  f  of  a?  =  30  /     |  of  a  =  75 

372.  How  long  is  a  line  f  of  which  is  9  in.  long  ? 

373.  A  man  rode  16  mi.  on  Monday,  which  was  f  of  the  dis- 
tance he  rode  on  Tuesday.     How  far  did  he  ride  on  Tuesday  ? 

374.  f  of  John's  money  is  210.     How  much  will  he  have 
left  if  he  gives  away  $  .05  ? 

375.  f  of  James's  money  is  $20.     How  much  will  he  have 
left  if  he  gives  away  £  of  his  money  ? 


DIVISION  OF  FRACTIONS  141 

376.  How  much  money  must  a  boy  have  that  he  may  lose  | 
of  it  and  have  12^  left?    30^?     42^?     60^? 

377.  How  much  money  must  a  man  have  so  that  after  gain- 
ing i  as  much,  he  may  have  $  700  ?     $  210  ?     $  441  ?     $  7.70  ? 

378.  Mary  has  12^.      Her  money  equals  y3^  of  Florence's 
money.     How  much  has  Florence  ? 

379.  William  has  8  marbles.     He  has  f  as  many  marbles  as 
James  has.     How  many  marbles  has  James  ? 

380.  Alice  has  14^,  which  is  just  -f  of  the  money  she  needs 
to  buy  her  geography.     What  is  the  price  of  the  geography  ? 

381.  Make  problems  in  which  a  certain  number  is  a  fractional 
part  of  the  number  which  is  to  be  found. 

382.  John  gave  away  -§-  of  his  marbles  and  then  had  30 
marbles  left.     How  many  had  he  at  first? 

383.  Susie  gave  -|  of  her  money  to  her  sister,  and  found  that 
she  had  16  cents  left.     How  much  had  she  at  first  ? 

384.  Harry  gave  away  -|  of  his  pigeons  and  sold  f  of  them. 
He  had  15  pigeons  left.     How  many  had  he  at  first  ? 

385.  In  a  storm,  a  ship's  crew  threw  overboard  30  bbl.  flour, 
which  was  y\  of  the   whole   cargo.       How  much   was  the 
whole  ? 

386.  A  man  owned  f  of  a  mine.      He  sold  J  of  his  share  for 
$  6000.     How  much  was  the  whole  mine  worth  ? 

387.  Mr.  Buchanan  sold  J  of  his  share  of  a  store  for  $  2000. 
What  was  his  share  worth?      His  share  was  f  of  the  whole 
value.     What  was  the  whole  value  ? 

388.  Owning  |  of  a  quarry,  Mr.  Harris  sold  ^  of  his  share 
for  $  6000.     What  was  the  value  of  the  quarry  ? 


142  FRACTIONS 

389.  Mr.  Madison  owned  \  of  an  Indiana  gas  well.     He  sold 
f  of  his  share  for  $  1500.      What  was  the  value  of  the  whole 
well? 

390.  yL  of  the  pins  in  a  cushion  were  crooked,  and  there  were 
66  straight  pins.     How  many  were  there  in  all,  and  how  many 
were  crooked  ? 

391.  If  8f  yd.  of  tape  cost  $  .70,  how  much  will  1  yd.  cost? 
9|yd.? 

392.  If  16f  yd.  of  rope  cost  100  cents,  how  much  will  1  yd. 
cost  ?     23f  yd.  ? 

393.  If  6  bu.  of  seed  cost  $15,  how  much  will  19f  bu.  cost? 

394.  If  |  of  a  quart  of  seed  cost  $  .18,  how  much  will  1  pk. 
cost? 

MISCELLANEOUS  EXERCISES 

1.  Divide  .0096  by  .12.    By  .008.    By  24.    By  3.2.    By  .16. 

2.  Divide  .000048  by  .012.     By  .4.     By  .0024.     By  2.4. 

3.  Divide  .144  by  .04.     By  48.     By  1.6.     By  .0003. 

4.  What  is  the  ratio  of  889.44  to  .102  ?    To  .105?    To  .108? 

5.  Multiply  7*  by  the  4th  prime.     92  by  the  6th  prime. 

6.  f  of  -f-g  of  y\  of  24  hr.  is  how  much  less  than  a  day  ? 

7.  24  sheets  of  paper  make  a  quire.     How  many  sheets  in 
|  of  |  of  |  of  a  quire  ? 

8.  How  many  sheets  in  y7^  of  ^  of  -f  of  a  quire  ?     In  |-  of 
£  of  -fe  of  a  quire  ? 

9.  Mrs.  Smith  is  f  as  old  as  Mr.  Smith,  who  is  48  yr.  old. 
Their  daughter  Alice  is  %  as  old  as  her  mother.     How  old  is 
Alice  ? 

10.  Add  yV,  &,  A,  iV    Add  A»  A>  A>  A- 

11.  From     L-  take      --.     From  -      take       . 


MISCELLANEOUS   EXERCISES  143 

•    12.  Multiply  the  first  prime  number  after  9  by  -|. 

13.  Multiply  the  largest  prime  factor  of  330  by  2^-. 

14.  Multiply  64£  by  the  largest  prime  factor  of  390. 

15.  Multiply  8f  by  the  1.  c.  m.  of  5,  6,  and  10. 

16.  Multiply  the  1.  c.  m.  of  8,  6,  9,  and  12  by  3^. 

17.  Multiply  the  g.  c.  d.  of  36,  48,  and  60  by  3|. 

Leto  =  f;  &  =  fi;  c  =  10£;  d  =  5£;  e  =  f    Find  values  of : 

18.  a  x  b  or  ab      ac      ad       ae       be       bd       cd       ce       de 

19.  a-f-c        a  +  e        a  +  d        b  +  c        b  +  a        d  +  e 

20.  c  — a        d  —  a        e  —  a         c  —  b        c  —  d        d  —  b 

Let  a  =  |;  &  =  3J;  c  =  l^;  d  =  2f;  e  =  f    Find  values  of : 

^      a         a         a         a         b          b          6cd 
6cdeeda6e 

22.  f  of  Anna's  money  is  $  .50.     How  much  will  she  have 
after  giving  away  7  f  ? 

23.  Of  what  number  is  21  three  fourths  ?     f  ?     f  ?     ^  ? 

24.  Of  what  number  is  16  four  sevenths  ?     |  ?     f  ?     ^  ? 

25.  If  ^  of  the  price  of  a  house  is  $  400,  what  is  the  price 
of  the  house  ?     How  much  will  five  such  houses  cost  ? 

26.  If  1  apple  costs  |^,  how  much  will  4  doz.  apples  cost  ? 

27.  If  f  of  the  price  of  an  orange  is  3^,  how  much  will  a 
dozen  oranges  cost  ?     6  doz.  ? 

28.  At   16|^   a  yard,  how  many  yards  of  ribbon  can  be 
bought  for$l?    $2?    $3?    $5? 

29.  12  doz.  make  a  gross.    When  buttons  are  bought  for  25 1 
a  gross,  what  is  the  cost  of  1  button  ? 

30.  If  500  pins  cost  10  ^,  how  much  will  1  pin  cost  ? 


144  FRACTIONS 

31.  If  a  gross  of  pencils  cost  50^,  how  much  will  1  pencil, 
cost? 

32.  Mrs.  Norton  paid  5^  for  a  box  of  toothpicks,  in  which 
there  were  2000  toothpicks.     How  many  did  she  get  for  a  cent  ? 
What  was  the  cost  of  1  toothpick  ? 

33.  She  paid  a  nickel  for  a  box  of  matches.     What  was  the 
price  of  each  match  if  there  were  500  matches  in  a  box  ? 

34.  A  gross  of  glass  vials  cost  48  ^.     How  much  did  1  vial 
cost  ? 

35.  Find  the  average  receipts  of  a  peanut  stand  for  6  days. 
Monday,  $  1.37 ;  Tuesday,  $  2.11 ;  Wednesday,  $  1.87 ;  Thurs- 
day, $  1.04  ;  Friday,  $  1.75 ;  Saturday,  9  3.10. 

36.  If  the  average  cost  of  keeping  up  the  stand  was  $  1.25 
per  day,  what  were  the  owner's  average  gains  per  day  ? 

37.  Suppose  a  pie  to  be  exactly  round,  and  101  in.  in  diam- 
eter.    If  it  were  cut  into  6  equal  pieces,  how  long  would  the 
curved  edge  of  each  piece  be  ? 

38.  The  surface  which  is  bounded  by 
an  arc  and  two   radii   is  called  a  Sector. 
Show  five  sectors  in  Fig.  4. 

You  may  remember  the  figure  of  a  sector 
more  easily  if  you  recall  the  way  in  which  pies, 
waffles,  and  round  cakes  are  usually  cut. 

39.  How  long  is  the  perimeter  of  a  sec- 
tor of  a  circle  whose  radius  is  4  in.,  if  the 

arc  of  the  sector  is  5^  in.  ?     Represent. 

40.  Draw  a  circle  whose  radius  is  3£  in.     Divide  it  into  4 
equal  sectors.     Write  the  word  "  sector  "  in  each.    Write  upon 
each  line  of  the  perimeter  of  a  sector  the  length  of  the  line 
and  find  the  length  of  the  perimeter  of  a  sector. 

41.  What  kind  of  an  angle  is  the  angle  of  a  sector  which  is 
J  of  a  circle  ?     Less  than  \  ?     Greater  than  1  ? 


MISCELLANEOUS   EXERCISES  145 

42.  Draw  a  circle  and  apply  its  radius  six  times  as  a  chord. 
What  regular  polygon  have  you  drawn  ?    Each  arc  thus  cut  off 
is  what  part  of  the  circumference?     If  the  radius  is  2f  in., 
how  long  is  the  circumference  ?    Each  arc  ? 

43.  Draw  lines  from  the  ends  of  each  arc  to  the  center  of  the 
circle.     What  are  these  lines  called  ?     What  kind  of  angles  do 
they  make  ? 

44.  Erase  the  chords.     Find  length  of  the  perimeter  of  each 
sector,  supposing  the  radius  of  the  circle  to  be  2-f  in.     5^  in. 
10£  in. 

45.  Erase  radii  so  as  to  leave  the  circle  divided  into  three 
equal  sectors.     Find  length  of  perimeter  of  each  sector,  assum- 
ing the  radius  to  be  -j-  in.    T9T  in.    lT^r  in. 

46.  If  the  radius  of  a  circle  is  2f  in.,  how  long  is  the  perim- 
eter of  a  sector  which  is  J  of  the  circle  ? 

47.  Find  the  perimeter  of  the  sector  which  remains  when 
a  sector  that  is  £  of  a  circle  is  subtracted  from  the  circle. 

48.  If  a  strip  of  paper  5  in.  long  were  curled  around  so  that 
its  edge  inclosed  a  circle,  how  long  would  the  circumference  of 
that  circle  be  ? 

49.  The  circumference  of  the  wheel  of  a  toy  wagon  is  20  in. 
How  far  does  the  wagon  run  when  the  wheel  turns  around 
once  ?     3  times  ? 

SUGGESTION.     Let  pupils  roll  a  coin,  button,  or  other  circular  objects, 
as  a  help  in  realizing  the  conditions  of  these  problems. 

50.  How  far  will  a  hoop    2^  ft.  in   circumference  run  in 
turning  7  times  ?     9  times  ? 

51.  How  many  times  will  a  wheel  2  ft.  in  circumference 
revolve  in  running  8  ft.? 

52.  How  many  times  will  a  wheel  6  ft.  in   circumference 
revolve  in  running  12  ft.?     8  yd.? 

HORN.    GRAM.    SCH.    AR.  — 10 


146  FRACTIONS 

53.  A  mile  is  5280  ft.     If  the  front  wheels  of  a  wagon  are 
each  6  ft.  in  circumference,  and  the  hind  wheels  are  8  ft.,  how 
many  times  will  each  wheel  revolve  in  running  a  mile  ? 

54.  Draw  a  rhomboid  whose  long  sides  are  each  double  a 
short  side.     How  long  would  its  perimeter  be  if  each  short 
side  were  8£  in.?     12  J  in.? 

55.  How  many  square  feet  in  a  lot  30  ft.  wide,  and  150  ft. 
deep  ?     If  the  owner  uses  \  of  the  lot  for  a  house,  and  1  for  a 
chicken  yard,  how  many  square  feet  remain  ? 

56.  A  house  is  48  ft.  long,  and  the  distance  from  the  ridge- 
pole to  the  eaves  on  each  side  is  23  ft.     How  many  shingles 
will  be  required  to  cover  it  if  6  shingles  are  required  to  cover  a 
square  foot? 

57.  John  is  8^  yr.  old,  and  his  sister  is  6T7T  yr.  old.     What 
is  their  average  age  ? 

58.  A  grocer  bought  3  cheeses,  one  weighing  32|-  lb.,  another 
28J  lb.,  another  41|  lb.     How  many  pounds  were  there  in  all  ? 

59.  Which  is  greater,  and  how  much,  -f-  x  f ,  or  -f-  -=-  J  ? 

60.  What  is  the  area  of  a  rectangle  3f  ft.  long  and  If  ft. 
wide  ? 

61.  What  is  the  area  of  a  right  triangle  whose  base  is  4£ 
in.,  and  altitude  3£  in.? 

62.  What  is  the  use  of  reducing  fractions  to  a  least  common 
denominator  ? 

63.  Write  the  fraction  that  expresses  the  ratio  of  the  first 
composite  number  after  18  to  the  first  composite  number  after 
30,  and  reduce  the  fraction  to  its  lowest  terms. 

64.  When  Arthur  was  a  year  old  his  father  placed  $50 
in  the  bank  as  money  to  be  used  in  sending  him  to  college. 
He  put  $  50  in  the  bank  on  every  birthday  until,  at  the  age 


MISCELLANEOUS  EXERCISES  147 

of  18,  Arthur  was  ready  for  college.     How  much  money  had 
been  placed  in  the  bank  for  him  ? 

65.  Arthur's   expenses  at  college  for  the  first  year  were 
$218.75;   for  the  second  year,  $310.50;   for  the  third  year, 
$  365.25.     How  much  of  the  amount  was  left  at  the  end  of  the 
third  year  ? 

66.  Arthur's  expenses  for  the  last  year  were  $  410.90.     He 
received   $465.67  as  interest.     How  much   was   left   of  the 
money  when  he  had  finished  his  college  course  ? 

67.  A  man  bought  a  lot  for  $  2000,  built  a  house  upon  it  for 
$  2500,  and  sold  the  property  so  as  to  gain  $  100  on  his  invest- 
ment.    For  how  much  did  he  sell  it  ? 

68.  Charles  bought  a  ball  for  $.08,  and  sold  it  for  $.12. 
The  gain  equaled  what  part  of  the  cost  ?     What  per  cent  ? 

69.  A  man's  salary  is  $  2400  a  year.     He  saves  J  of  it  one 
year,  1  of  it  the  next  yea^  and  -|  of  it  the  next  year.     How 
much  has  he  saved  at  the  end  of  the  third  year  ? 

70.  A  gentleman  had  1200  books  in  his  library,  and  gave 
away  -§•  of  them.     He  lost  -^  of  the  remainder.     How  many 
books  were  left  ? 

71.  He  added  200  more  volumes  to  the  library,  and  then 
gave  away  f  of  it.     How  many  had  he  left  ? 

72.  The  base  of  an  isosceles  triangle  is  3  ft.     The  ratio 
of  one  of  the  equal  sides  to  the  base  is  -J.     How  long  is  the 
perimeter  of  the  triangle  ?     Kepresent. 

73.  How  long  is  the  perimeter  of  an  isosceles  triangle  whose 
base  is  14  in.  and  each  of  whose  equal  sides  is  5  in.  longer  than 
the  base  ? 

74.  How  long  is  the  perimeter  of  an  isosceles  triangle  whose 
base  is  21  in.  and  each  of  whose  equal  sides  is  33|%  longer 
than  the  base  ? 


148 


FRACTIONS 


75.  The  perimeter  of  a  certain  isosceles  triangle  is  25  in. 
and  one  of  the  equal  sides  is  9  in.     How  long  is  the  base  ? 

76.  The   base  of   an   isosceles   triangle  is  11  in.  and  the 
perimeter  35  in.     How  long  is  each  of  the  equal  sides  ? 


FIG.  5. 


77.  Draw  a  rectangle  4  in.  long  and  3 
in.  wide.  Draw  a  diagonal  of  it.  Into 
what  kind  of  figures  does  a  diagonal 
divide  a  rectangle?  If  the  angles  of 
your  figure  are  exact  right  angles,  and  if 
your  lines  are  exactly  drawn,  the  diag- 
onal will  be  just  5  in.  long. 

SUGGESTION  TO  TEACHER.  Let  pupils  find  by  trial  that  if  3  in.  be 
measured  off  upon  one  of  the  lines  about  a  right  angle  and  4  in.  upon  the 
other  line,  the  joining  line  will  be  5  in.  long. 

A  78.    In  the  right  triangle  ABC  how  long  is 

the  hypotenuse  AC  if  the  numbers  represent 
inches  ? 

79.  If  BC  and  AB  were  each  twice  as  long 
as  they  are,  AC  would  be  twice  as  long  as  it 
is.     If  AB  is  8  in.  and  BC  6  in.,  how  long  is 
AC?     Prove  by  measuring. 

80.  If  the  perpendicular  sides  of  a  right 
triangle  are  in  the  ratio  of  3  to  4,  the  ratio 

of  the  hypotenuse  to  the  less  side  is  f,  and  the  ratio  of  the 
hypotenuse  to  the  other  side  is  f .  In  a  right  triangle  whose 
base  is  30  and  altitude  40  how  long  is  the  hypotenuse? 
Represent. 

81.  How  long  is  the  hypotenuse  of  a  -right  triangle  whose 
perpendicular  sides  are  9  in.  and  12  in.  ?     21  and  28  ?     15  and 
20  ?     33  and  44  ? 

82.  The   first   steamship   crossed    the    Atlantic    Ocean   in 
MDCCCXIX.     Tor  how  many  years  has  it  been  possible  for 
Americans  to  go  to  Europe  in  a  steamship  ? 


3 

FIG.  6. 


MISCELLANEOUS   EXERCISES  149 

83.  Imagine  a  block  of  ice  1  yd.  in  dimensions.     How  many 
square  feet  are  there  in  all  the  surfaces  ? 

84.  Imagine  the  same  figure  with  one  cubic  foot  cut  out  of 
one  corner  of  it.     How  many  square  feet  in  all  its  surfaces  ? 

85.  Imagine  a  cubic  yard  of  ice,  and  suppose  a  cubic  foot  of 
it  to  be  cut  from  the  middle  of  one  side.     How  many  square 
feet  in  all  the  surfaces  of  the  solid  that  is  left  ? 

86.  How  many  square  feet  in  all  the  surfaces  of  the  solid 
that  would  be  left,  if  the  cubic  foot  were  put  back  in  its  place 
and  the  cubic  foot  above  it  were  taken  away  ? 

87.  If  a  box  1  yd.  in  dimensions  were  packed  f  full  of 
groceries,  how  many  cubic  feet  of  space  would  be  left  ? 

88.  What  part  of  a  cubic  yard  is  a  cube  which  is  f  of  a  yard 
in  dimensions  ?     How  many  cubic  feet  are  there  in  it  ? 

89.  A  coal  dealer  bought  1246  tons  of  coal  at  $  4J-  a  ton,  and 
sold  it  for  $  6£  a  ton.     What  was  his  gain  on  each  ton  ?     On 
the  whole  ? 

90.  A  man  bought   $  88^-  worth   of  furniture,  paying  in 
weekly  installments  of  $  14^  each.     In  how  many  weeks  did 
he  pay  for  the  furniture  ? 

91.  A  grocer  bought  strawberries  at  the  rate  of  4  boxes  for 
a  quarter,  and  sold  them,  at  the  rate  of  3  boxes  for  a  quarter. 
How  much  did  he  gain  on  each  box  ?     On  a  dozen  boxes  ?    On 
a  gross  of  boxes  ? 

92.  Mr.  Jones  worked  |-  of  a  day  on  Monday,  f  of  a  day  on 
Tuesday,  and  a  whole  day  on  Wednesday,  on  Thursday,  and 
on  Friday.     On  Saturday  he  worked  %  a  day.     At  $  3  per  day, 
how  much  did  he  earn  in  the  week  ? 


CHAPTER   V 

DENOMINATE   NUMBERS 

1.  How  many  feet  equal  a  yard?     How  many  pints  equal 
a  quart?      How  many  ounces  equal  a  pound?.  How  was  it 
decided  in  these  cases  how  many  units  of  a  certain  denomina- 
tion should  make  one  of  the  next  higher  denomination? 

SUGGESTION  TO  TEACHER.  Let  facts  concerning  the  origin  of  our  systems 
of  measuring  be  obtained  from  encyclopedias  and  other  sources  of  informa- 
tion and  brought  to  the  class.  Pupils  should  understand  that  the  value  of 
a  unit  in  terms  of  lower  denominations  is  an  arbitrary  value,  varying  in 
different  kinds  of  measurements. 

2.  Numbers  that  show  measurements  whose  values  are  set- 
tled by  custom  or  law  are  called  Denominate  Numbers,  as  5 
bushels,  2  hours,  1  dollar.     Denominate  numbers  that  consist 
of  more  than  one  denomination  are  called  Compound  Denominate 
Numbers.     Write  a  compound  denominate  number  whose  larg- 
est denomination  is  bushels.     Hours.     Tons.     Miles.     Acres. 
Gallons.     Dollars.     Meters. 

3.  The  denominations  of  United  States  money  are  mills  (m.), 
cents  (£),  dimes  (d.),  dollars  ($)  and  Eagles  (E.). 

UNITED  STATES  MONEY 
One  dollar  is  the  standard 
1  eagle  =  10  dollars 
1  dollar 

1  dime  =  ^  of  a  dollar 
1  cent  =  jfa  of  a  dollar 

1  mill  =  j^Vo  of  a  dollar 
150 


DENOMINATE   NUMBERS  151 

4.  Name   each   denomination  of  the   following :    $  5875. 
$10,125.     $20,705. 

5.  Express  3  dollars  as  cents.     As  dimes.     As  eagles. 

6.  A  10-dollar  gold  piece  is  called  an  Eagle.     A  20-dollar 
gold  piece  is  called  a  Double  Eagle.      A  5-dollar  gold  piece  is 
called  a  Half  Eagle.     What  is  the  value  of  a  Quarter  Eagle  ? 

7.  Name  the  silver  coins.     What  other  coins  are  there? 

8.  How  much  money  has  a  man  who  has  2  double  eagles,  an 
eagle,   3  half  eagles,  a  quarter  eagle,  2  dollars,  3  dimes,  2 
nickels,  and  3  cents  ? 

9.  CLASS  EXERCISE.     may  name  a  certain  number  of 

different  kinds  of  coins,  and  the  class  may  find  the  amount 
of  money  which  their  sum  equals. 

10.  Treasury  or  bank  notes  are  also  used  as  money.     If  a 
man  has  eight  $  100  bills,  seven  $  20  bills,  a  $  10  bill,  a  $  5 
bill,  a  $  2  bill,  and  three  $  1  bills,  how  much  less  than  $  1000 
has  he  ? 

11.  How  many  mills  in  a  dollar?     In  a  half  eagle?    Why 
is  there  no  coin  to  represent  a  mill  ? 

12.  The  denominations  of  liquid  measure  are  gills  (gi.), 
pints  (pt.),  quarts  (qt.)  and  gallons  (gal.). 

LIQUID  MEASURE 
424 
gal.        qt.       pt.       gi. 

Over  the  abbreviation  of  each  denomination  above  you  will  find  the 
number  of  units  that  equal  a  unit  of  tha.next  higher  denomination. 

Give  the  table  of  liquid  measure,  beginning  with  the  units 
of  the  lowest  denomination. 

13.  Fill  the  blank  in  the  following  table  of  equivalent  values : 

1  gal.  =  4  qt.  =  8  pt.  =  —  gi. 

SUGGESTION  TO  TEACHER.  Pupils  should  make  actual  measurements,  so 
far  as  is  practicable,  in  connection  with  the  study  of  each  table,  and  should 
learn  to  change  rapidly  units  of  one  denomination  into  units  of  another. 


152  DENOMINATE   NUMBERS 

14.  Illustrate  each  of  the  following  statements : 

a  As  4  gills  equal  a  pint,  any  number  of   pints  equals  4 
times  as  many  gills  as  pints. 

b    As  2  pints  equal  a  quart,  any  number  of  quarts  equals 
twice  as  many  pints  as  quarts. 

c    As  4  quarts  equal  a  gallon,  any  number  of  gallons  equals 
4  times  as  many  quarts  as  gallons. 

15.  Express  -f  gal.  as  quarts,     £  qt.  as  pints,     f  pt.  as  gills. 

16.  Express  %  gal.  as  quarts,    -§-  qt.  as  pints.    T8T  pt.  as  gills. 

17.  Express  .75  gal.  as  quarts.     .5  qt.  as  pints.     .625  pt.  as 
gills. 

18.  Express  .375  gal.  as  quarts.     As  pints. 

19.  Express  5|-  gal.  as  quarts.     As  pints.     As  gills. 

20.  Express  1\  gal.  as  pints. 

21.  Express  2|  gal.  as  gills. 

22.  Express  3^  gal.  as  quarts.     3  gal.  and  2  qt.  as  quarts. 

23.  How  many  quarts  in  5  gal.  2  qt.  ?     7  gal.  1  qt.  ? 

24.  How  many  pints  in  7  qt.  3  pt.  ?     In  1  gal.  3  pt.  ? 

25.  How  many  gills  in  1  pt.  3  gi.  ?    1  qt.  3  gi.  ?    3  qt.  1  gi.  ? 

26.  1  gi.  equals  what  part  of  1  qt.  1  pt.  ? 

1  qt.  1  pt.  =  5  pt.,  which  equal  20  gi. 
1  gi.  =  Js  of  20  gi. 

27.  1  gi.  equals  what  part  of  2  qt.  1  pt.  ?     Of  3  qt.  1  pt.  ? 
Of  1  gal.  1  pt.  ? 

28.  Illustrate  each  of  the  following  statements : 

a   As  4  gills  make  a  pint,  any  number  of  gills  equals  \  as 
many  pints  as  gills. 


DENOMINATE   NUMBEKS  153 

b  As  2  pints  equal  1  quart,  any  number  of  pints  equals  £  as 
many  quarts  as  pints. 

c  As  4  quarts  equal  1  gallon,  any  number  of  quarts  equals  £ 
as  many  gallons  as  quarts. 

29.  Express  32  gi.  as  pints.     As  quarts.     As  gallons. 

30.  Express  40  pt.  as  quarts.     As  gallons. 

31.  Express  25  gi.  as  pints.     Ans.  6|  pt. 

32.  Express  7  pt.  as  quarts.     9  qt.  as  gallons. 

33.  Express  11  pt.  as  quarts.     As  gallons. 

34.  Express  13  pt.  as  quarts  and  pints.     Ans.  6  qt.  1  pt. 

35.  Express  15^  pt.  as  quarts,  pints,  and  gills. 

36.  Express  17^  qt.  as  gallons,  quarts,  and  pints. 

37.  Express  19£  pt.  as  gallons,  quarts,  pints,  and  gills. 

38.  1  gi.  equals  what  part  of  a  pint  ?     Of  a  quart  ?     Of  a 
gallon  ? 

39.  3  gi.  equal  what  part  of  a  quart  ?     Of  a  gallon  ? 

40.  Express  If  pt.  as  quarts.     As  gallons. 

41.  Express  1  pt.  3  gi.  as  pints. 

42.  Express  2  qt.  1  pt.  2  gi.  as  pints.    As  quarts.    As  gallons. 

43.  Which  is  greater  and  how  much,  2  gal.  1  qt.  3  pt.  or 
22  pt.  ? 

44.  Express  1  pt.  as  a  decimal  of  a  quart.      3  qt.  as  a  deci- 
mal of  a  gallon. 

45.  At  6£  cents  a  quart,  how  much  will  a  gallon  of  cider 
cost  ?     31  gal.  ?     4f  gal.  ?     1  gal.  3  qt.  ?     1  pt.  ?     3  qt.  1  pt.  ? 

46.  Name   several   articles   that  are   measured    by  liquid 
measure. 


154  DENOMINATE   NUMBERS 

47.  Add: 

gal   qt    pt       i  We  find  the  sum  of  the  gills  to  be  7  git     7  &*• 

equal  1  pt.  3  gi.     We  place  the  3  gi.  under  the  col- 

2312  urnn  of  gills  and  add  the  1  pt.  to  the  number  of  pints. 
7103  The  sum  of  the  pints  is  3  pt.,  equal  to  1  qt.  1  pt. 
4212  The  *  pt>  is  P^aced  under  the  column  of  pints,  and  the 

1  qt.  is  added  to  the  number  of  quarts.     The  sum  of 

14     3     1     3        the  quarts  is  7  qt.,  equal  to  1  gal.  3  qt.     The  3  qt. 
are  placed  under  the  column  of  quarts,  and  the  1  gal. 
is  added  to  the  number  of  gallons,  making  14  gal.  3  qt.  1  pt.  3  gi. 

Add: 

gal.  qt.    pt.    gi.                    gal.  qt.    pt.    gi.  gal.  qt.   pt.    gi. 

48.  1  3  1  3  49.   5  2  1  1  50.   8  3  0  3 
3302       7313  9302 

gal.  qt.  pt.  gi.        gal.  qt.  pt.  gi.        gal.  qt.  pt.  gi. 

51.  6  3  1  2    52.  11  3  1  1    53.  15  2  1  1 

7213       5213       11  3  1  3 

54.  A  milkman  leaves  25  gal.  3  qt.  1  pt.  of  milk  at  one 
hotel,  and  33  gal.  2  qt.  1  pt.  at  another.     How  much  does  he 
leave  at  both  hotels  ? 

gal.  qt.   pt.    gi. 

55.  From  7313  Subtract  each  number  in  the  subtra- 
,   -i         £»     1      1      2      nend  from  the  corresponding  number  in 

. the  minuend. 

2201 

56.  CLASS  EXERCISE.     may  write  a  compound  denomi- 
nate number  consisting  of  gal.,  qt.,  pt.,  and  gi.     The  class  may 
use  it  as  a  minuend,  making  every  number  in  the  subtrahend 
less  than  its  corresponding  term  in  the  minuend. 

gal.  qt.  pt.   gi.          As  3  gi.  cannot  be  taken  from  1  gi., 

57.  From  7111      we  reduce  *  P*-  *  S1-  to  Sills>  which  gives 


take     2303 


5  gi.     5  gi.  minus  3  gi.  equal  2  gi.     As 
the  1  pt.  has  been  taken  from  the  column 
4202       of  Pints  and  reduced  to  gills,  there  are  no 
pints  left  in  the  minuend,  from  which 

0  pt.  are  to  be  taken.     As  3  qt.  cannot  be  taken  from  1  qt.,  we  reduce 

1  gal.  to  quarts,  which,  with  the  1  qt.,  equal  5  qt.     3  qt.  from  5  qt.  leave 

2  qt.     As  1  gal.  has  been  taken  from  the  column  of  gallons  and  reduced 
to  quarts,  only  6  gal.  remain.     6  gal.  minus  2  gal.  equal  4  gal.     Hence  the 
difference  is  4  gal.  2  qt.  0  pt.  2  gi. 


DENOMINATE   NUMBERS  155 

Subtract : 

gal.  qt.  pt.  gi.  gal.  qt.  pt.  gi. 

58.  17  3  0  3  59.  16  3  1  1 

11  111  8103 


gal.  qt.  pt.  gi.  gal.  qt.  pt.  gi. 

60.  13  1  1  2  61.  15  2  1  0 

6311  9303 


62.  17  gal.  1  qt.  1  pt.  of  oil  were  in  a  tank.     11  gal.  2  qt. 
1  pt.  were  drawn  out.     How  much  remained  ? 

63.  Multiply  3  gal.  1  qt.  1  pt.  2  gi.  by  9. 

gal.  qt.   pt.    gi.          9  times  2  gi.  equal  18  gi.,  which  equal  4  pt.  2  gi. 

3112       ^e  ^  &*•  are  wr^ten  under  the  gills.     9  times  1  pt. 

p      plus   the  4  pt.  already  found  equal  13  pt.,  which 

equal  6  qt.  1  pt.     9  times  1  qt.  plus  the  6  qt.  already 

30     3     J     2      found  equal  15  qt.     15  qt.  equal  3  gal.  3  qt.     9  times 

3  gal.  plus  the  3  gal.  already  found  equal  30  gal. 

Hence  the  product  is  30  gal.  3  qt.  1  pt.  2  gi. 

gal.     qt.      pt.      gi.  gal.      qt.      pt.      gi 

64.  Multiply  5312             65.    10       113 
by  6  9_ 

66.  Multiply  7  gal.  2  qt.  1  pt.  3  gi.  by  3.     By  5.     By  7. 

67.  A  milkman  sold  99  gal.  3  qt.  of  milk  on  Monday.     If 
he  were  to  sell  the  same  amount  every  day  for  a  week,  how 
much  milk  would  he  sell  ? 

68.  At  24^  a  gallon,  how  much  would  he  receive  for  the 
milk  ?     If  the  whole  cost  of  the  milk  was  $  135,  how  much 
would  he  gain  ? 

69.  How  many  times  is  3  gal.  3  qt.  1  pt.  contained  in  19  gal. 
1  qt.  1  pt.  ? 

Express  both  dividend  and  divisor  in  the  same  denomination  before 
dividing. 


156  DENOMINATE  NUMBERS 

70.  How  many  bottles  each  containing  1  pt.  2  gi.  can  be 
filled  from  a  flask  containing  3  gal.  ? 

71.  How  many  bottles  each  containing  1  pt.  2  gi.  can  be 
filled  from  a  6-gallon  tank  ? 

72.  Divide  21  gal.  3  qt.  1  pt.  3  gi.  by  6. 

In  dividing  denominate  numbers,  if  there  is  a  remainder  after  dividing, 
it  is  the  custom  to  reduce  that  remainder  to  the  next  lower  denomination 
instead  of  writing  the  quotient  as  a  mixed  number.  In  this  way,  frac- 
tions are  avoided  in  all  the  denominations  of  the  quotient  except  the 
lowest. 

Dividing  21  gal.  by  6,  we  have  3  gal.  for  the  quotient  and  3  gal.  for  the 

remainder.     3  gal.  or  12  qt.  plus  3  qt.  equal  15  qt.     15  qt.  divided  by  6 

give  2  qt.  for  a  quotient  and  3  qt.  for  a  remainder. 

gal.   qt.  pt.     gi.       3  ctf"  or  6  Pk  Plus  !  Pfc-  e(Jual  7  P*-     7  ?*•  divided 

6 1   21     3     1     3        ky  6  give  1  pt'  for  a  ^uotient  and   1  P*-  for  a 
' — - — - — - — —      remainder.     1  pt.  or  4  gi.  plus  3  gi.  equal  7  gi. 

^"      7  gi.  divided  by  6  equal  1£  gi.    Hence  the  quotient 
is  3  gal.  2  qt,  1  pt.  1£  gi. 

gal.     qt.      pt.      gi.  gal.     qt.      pt.      g 

73.  Divide  2)8      2      1      2  74.   3)6      3      1      2 

75.  Divide  9  gal.  3  qt.  1  pt.  1  gi.  by  4.    By  5.    By  6.    By  8. 

76.  How  many  half-pint  bottles  can  be  filled  from  a  10- 
gallon  can  of  milk? 

77-  The  denominations  of  dry  measure  are  pints  (pt.), 
quarts  (qt.),  pecks  (pk.),  and  bushels  (bu.). 

DRY  MEASURE 

482 
bu.      pk.      qt.      pt. 

Give  the  table  of  dry  measure,  beginning  with  the  units  of 
the  lowest  denomination. 

78.  Fill  the  blanks:  1  bu.  = — pk.  = — qt.  =— pt. 

79.  Give  the  ratio  of  1  pt.  to  a  unit  of  each  denomination 
of  dry  measure. 


DENOMINATE   NUMBERS  157 

80.  Express  3  qt.  1  pt.  as  pints.     As  quarts.     As  pecks. 

81.  Express  2  pk.  5  qt.  1  pt.  as  pt.     As  qt.     As  pk.     As  bu. 

82.  Name  several  articles  that  are  measured  by  dry  measure. 

83.  At  20^  a  peck,  how  much  does  a  bushel  of  tomatoes 
cost  ?     3  qt.  1  pt.  ?     5  qt.  1  pt.  ? 

84.  At  25^  a  quart,  how  much  does  a  pint  of  strawberries 
cost  ?     3  qt.  1  pt.  ?     5  qt.  1  pt.  ? 

85.  At  121  ^  a  quart,  how  much  does  a  peck  of  potatoes 
cost  ?     1  pk.  2  qt.  ?     1  bu.  ? 

86.  Express  -f  bu.  as  pk.   %  pk.  as  qt.    -J  qt.  as  pt.    f  bu.  as  qt. 

87.  Express  f  pk.  as  bu.   T8T  qt.  as  pk.  ^  pt.  as  qt.    |-  qt.  as  bu. 

88.  Express  .125  pk.  as  qt.     .875  bu.  as  pk.     55  qt.  as  pt. 

89.  Express  .375  bu.  as  pk.     As  qt.     As  pt. 

90.  Express  4  qt.  as  a  decimal  of  a  peck.     Of  a  bushel. 

91.  How  many  pints  in  -J  of  ^y  of  J  of  a  bushel  ? 

92.  By  selling  apples  at  f  .40  a  peck,  Mr.  Allen  doubled 
his  money.     How  much  did  they  cost  him  per  bushel  ? 

93.  Express  in  pecks  7%  of  a  bushel.     Express  in  quarts 
51  %  of  a  peck. 

Add: 

bu.      pk.      qt.      pt.  bu.      pk.      qt.      pt. 

94.  20       3       5       1  95.    21       3       1       1 

4111  33   201 

5261  48   371 


bu.   pk.  qt.   pt.  bu.   pk.   qt.   pt. 

96.   6   2   5   1  97.  10   3   6   1 

3340  8151 

6171  12   240 


158  DENOMINATE   NUMBEKS 
Subtract : 

bu.      pk.  qt.      pt.                                   bu.      pk.      qt.      pt. 

98.    18       3  2       1                      99.    40       1       5       0 

14       1  6       1                              17       3       2       1 


bu.      pk.     qt.      pt.  bu.      pk.      qt.      pt. 

100.    8360  101.   19      1       2      1 

1241  14      361 


Multiply : 

bu.      pk.     qt.      pt.  bu.      pk.      qt.      pt. 

102.    4261  103.      8371 

5  7 

bu.     pk.      qt.      pt.  bu.      pk.      qt.      pt. 

104.    7151  105.      8371 

6  8 


106.  Multiply  6  bu.  3  pk.  3  qt.  1  pt.  by  4.     By  6.     By  8. 

107.  A  grocer  has  3  bins,  each  holding  4  bu.  3  pk.  2  qt. 
of  potatoes.     How  much  do  they  all  hold  ? 

108.  How  much  wheat  is  there  in  10  bins,  if  each  bin  con- 
tains 40  bu.  2  pk.  6  qt.  ? 

109.  At  5 1  a  quart,  how  much  will  a  bushel  of  walnuts  cost  ? 

bu.    pk.     qt.     pt.  bu.    pk.     qt.     pt. 

110.  Divide    5)6      1      7      1  111.    6)8      2      5      1 

112.  Divide  9  bu.  2  pk.  7  qt.  1  pt.  by  2.     By  3.     By  4. 

113.  How  many  boxes  containing  2  bu.  3  pk.  of  sawdust 
can  be  emptied  into  a  bin  which  will  hold  13  bu.  3  pk.  ? 

114.  Place  a  cubic  centimeter  upon  each  of  the  corners  of 
the  upper  surface  of  a  cube  that  holds  a  liter,  and  find  how 
many  square  centimeters  there  are  in  the  surface  of  the  figure 
thus  formed. 


DENOMINATE   NUMBERS  159 

115.  The  standard  unit  of  metric  measure  of  capacity  is 
the  Liter,  equal  to  about  .9  of  a  quart  dry  measure  and  1.05 
quarts  liquid  measure. 

METRIC  MEASURE  OF  CAPACITY 

1  kiloliter  (Kl.)  =  1000  liters 
1  hectoliter  (HI.)  =  100  liters 

1  decaliter  (Dl.)  =  10  liters 
1  liter  (1.) 

1  deciliter  (dl.)  =  .1  of  a  liter 

1  centiliter  (cl.)  =  .01  of  a  liter 

1  milliliter  (ml.)  =  .001  of  a  liter 

Write  table  of  equivalents. 

1  Kl.  =  10  HI.  =  —  Dl.  =  —1.  =  —  dl.  =  —  cl.  =  —ml. 

116.  Learn  to  give  quickly,  forward  and  backward,  the  de- 
nominations of  this  table,  and  the  meaning  of  each  prefix  to 
the  word  "liter." 

To  help  you  remember  the  value  of  the  units  observe  that  D,  H,  and 
K,  the  abbreviations  for  Deca,  Hecto,  and  Kilo,  follow  one  another  in 
alphabetical  order. 

117.  Give  the  meaning  of   the  following   prefixes:     Kilo, 
milli,  Hecto,  centi,  Deca,  deci.     Of  d,  H,  D,  c,  K. 

118.  Bead  as  a  decimal  1235.576  1.     Give  the  denomination 
of  each  figure. 

119.  How  many  centiliters  in  5.37  1.  ?    In  8.4  1.  ?    10.251.? 
6.875  1.  ? 

120.  How  are  liters  reduced  to  centiliters  ? 

121.  Express   7   kiloliters   as   liters.     As   hectoliters.     As 
centiliters.     As  milliliters. 

122.  Express  2  HI.  5  Dl.'as  1.     As  cl.     As  Kl.     As  dl. 

123.  Reduce  2345.248  1.  to  units  of  each  of  the  other  de- 
nominations. 

124.  CLASS  EXERCISE.        —  may  name  a  number  of  liters, 
and  the  class  may  reduce  them  to  dl.     To  cl.     To  ml. 


160  DENOMINATE  NUMBERS 

125.  Write  in  one  number,  5  Kl.  2  HI.  5  Dl.  2  1.  0  dl.  7  ml. 

126.  Write  and  add :    3  Kl.  0  HI.  2  Dl.  5  1.  0  dl.  2  cl.  7  ml. 
2  HI.  4  Dl.  0  1.  2  dl.  7  cl.     9  Kl.  3  HI.  2  Dl.  6  1.  5  dl.  2  cl.  8  ml. 

127.  From  8  Kl.  2  HI.  7  Dl.  4  1.  6  dl.  2  cl.  9  ml. 
take    6  HI.  4  Dl.  2  1.  3  dl.  8  cl.  5  ml. 

128.  825.346  1.  -  27.59  1.  =  ? 

129.  Multiply  125.275  1.  by  5.     By  8.     By  12. 

130.  Multiply  341.626  1.  by  10.     By  100.     By  1000. 

131.  Divide  239.268  1.  by  4.     By  6.     By  12. 

132.  Under   which   system   of   denominate   numbers   is   it 
easier  to  add,  subtract,  multiply,  and  divide,  the  metric  system 
or  the  English  system  ?     Why  ? 

133.  How  many  liters  in  7%  of  132.5  1.  ?     Of  178.7  1.  ? 

134.  17  1.  +3%  of  17  1.  =  how  many  liters?     How  many 
dl.?    How  many  cl.? 

135.  How  much  will  7.5  1.  of  wine  cost  at  $  1.25  a  liter  ? 

136.  What  is  the  cost  of  a  Dl.  of  wheat,  at  $  7.25  per  HI.? 

137.  What  is  the  cost  of  a  hogshead  of  wine  containing 
225  1.  at  $  .15  per  liter  ? 

138.  What  is  the  cost  of  25  1.  of  vinegar  at  $  15  per  HI.? 

139.  How  much  wheat  is  contained  in  125  sacks,  each  hold- 
ing 1  HI.  2  Dl.  ? 

140.  The  denominations  of  avoirdupois  weight  are  ounces 
(oz.),  pounds  (lb.),  hundredweights  (cwt.),  and  tons  (T.). 

AVOIRDUPOIS  WEIGHT 

20  100  16 

T.  cwt.  lb.  oz. 

Fill  out  the  following  table  of  equivalents. 

1  T.  =  —  cwt.  =  —  lb.  =  —  oz. 


DENOMINATE   NUMBERS  161 

141.  1  oz.  is  what  part  of  a  pound  ?     Of  a  hundredweight  ? 
Of  a  ton  ? 

142.  Express  5  Ib.  8  oz.  as  ounces.     As  pounds.     As  hun- 
dredweights.    As  tons. 

143.  Express  3  T.  10  cwt.  25  Ib.  12  oz.  in  each  denomination 
of  avoirdupois  weight. 

144.  Express  -fa  T.  as  cwt.     f  cwt.  as  Ib. 

145.  Express  f  oz.  as  Ib.     -f  £  Ib.  as  cwt.     f§£  Ib.  as  cwt. 

146.  Express  .7  T.  as  cwt.     .17  cwt.  as  Ib.     .125  Ib.  as  oz. 
.75  T.  as  Ib. 

147.  Express  .625  T.  as  cwt.     As  Ib.     As  oz. 

148.  Express  15  Ib.  as  a  decimal  of  a  hundredweight. 

149.  At  20^  a  pound,  how  much  will  2  Ib.  and  8  oz.  of 
butter  cost  ?     3  Ib.  4  oz.  ?     5  Ib.  ?     12  oz.  ?     7  Ib.  2  oz.  ? 

150.  How  many  pounds  in  21  T.  ?     In  3%  of  a  ton  ? 

151.  A  farmer  brought  a  ton  of  hay  to  market  and  sold 
25%  of  it.     How  many  pounds  had  he  left  ? 

152.  How  many  cwt.  in  5%  of  a  ton  ?    In  45%  ? 

153.  1  Ib.  —  f  of  a  pound  =  how  many  ounces  ? 
Add: 


T. 

cwt. 

ib. 

oz. 

T. 

cwt. 

Ib. 

oz. 

154. 

7 

15 

75 

8 

155.  10 

19 

67 

5 

4 

12 

55 

12 

12 

14 

25 

13 

6 

17 

80 

15 

15 

16 

84 

11 

T. 

Ib. 

oz. 

T. 

Ib. 

oz. 

156. 

8 

425 

7 

157.  16 

875 

10 

9 

375 

8 

14 

985 

11 

7 

425 

5 

6 

435 

13 

HORN.    GRAM.    SCH.    AR. 11 


162  DENOMINATE   NUMBERS 

T.    cwt.       Ib.       oz.  T.     cwt.     Ib.      oz. 

158.    From     9       3       30       4  159.    10       0       5       0 

take       27      41       7  6728 


T. 

Ib. 

oz. 

T. 

Ib. 

oz. 

160. 

From 

8 

201 

8 

161.  18 

700 

12 

take 

4 

175 

12 

4 

900 

8 

162.  From  30  T.  800  Ib.  of  hay  there  were  sold  7  T.  and 
900  Ib.     How  much  was  left  ? 

163.  Mrs.  Harris  brought  20  Ib.  8  oz.  of  butter  to  market, 
and  sold  18  Ib.  and  12  oz.     How  much  remained  unsold  ? 

164.  Name  articles  that  are  weighed  by  avoirdupois  weight. 

165.  John  weighs  87£  Ib.,  Thomas  92£  Ib.,  William  97f  Ib. 
How  much  do  they  all  weigh  ?     Give  answer  in  pounds  and 
ounces. 

SUGGESTION  FOR  CLASS   EXERCISE.  Find  sums  and  differences  of 
weights  of  pupils. 

T.    cwt.     Ib.      oz.  T.     cwt.     Ib.     oz. 

166.  Multiply    5     10    40    10  167.    15     15     75     15 
by               8                       12 

168.  Multiply  15  T.  7  cwt.  25  Ib.  8  oz.  by  3.     By  5.     By  8. 
By  9. 

Divide : 

T.    cwt.     Ib.    oz.  T.     cwt.     Ib.     oz. 

169.  4)9    15    20    8  170.   5)16    14    50    10 

171.  Divide  20  T.  12  cwt.  48  Ib.  12  oz.  by  6.    By  8.    By  12. 

172.  One  Christmas  day  40  T.  of  coal  were  equally  distrib- 
uted among   11   poor  families.     How   many   tons,   hundred- 
weights, and  pounds  did  each  receive  ? 

173.  Formerly  2240  Ib.   were  considered  a  ton,  and  that 
standard  is  sometimes  used  now.     In  that  case  the  ton  was 
called  a  "long  ton."     How  many  pounds  in  4J  long  tons ?     In 

of  along  ton? 


DENOMINATE  NUMBERS  163 

174.  A  coal  dealer  buys  150  T.  of  coal,  2240  Ib.  each,  at 
$  4.50  per  ton.     He  sells  it  at  $  4.75  per  ton,  giving  2000  Ib. 
per  ton.     How  much  does  he  gain  ? 

175.  The  standard  unit  of  metric  measure  of  weight  is  a 
Gram,  equal  to  about  15^-  Troy  grains. 

METRIC  MEASURE  OF  WEIGHT 

1  kilogram  (Kg.)  =  1000  grams 

1  hectogram  (Hg.)  =  100  grams 

1  decagram  (Dg.)  =  10  grams 

1  gram  (g.) 

1  decigram  (dg.)  =  .1  of  a  gram 

1  centigram  (eg.)  =  .01  of  a  gram 

1  milligram  (mg.)  =  .001  of  a  gram 

Fill  blanks : 

1  Kg.  =  —  Hg.  =  —  Dg.  =  —  g.  =  —  dg.  =  —  eg.  =—  mg. 

176.  The  weight  of  a  cubic  centimeter  of  water  is  a  gram. 
How  many  grams  does  a  liter  of  water  weigh? 

177.  What  part  of  a  kilogram  is  a  decagram?    Decigram? 
Milligram  ?     Hectogram  ?     Centigram  ? 

178.  Name  each  denomination  of  the  expression  1978.347  g. 

179.  Write  in  one  number,  making  the  gram  the  unit  4  Kg. 
7  Hg.     6  Dg.     5  g.     2  dg.     6  eg.     3  mg. 

180.  Express  75  Kg.  as  grams.     As  eg.     As  mg.     As  Dg. 
As  dg.     As  Hg. 

181.  Express  186  eg.  as  grams.     As  mg.     As  Dg.     As  dg. 
As  Hg.     As  Kg. 

182.  Add  325  g.,  55  Kg.,  75  Dg. 

183.  How  many  grams  in  20%  of  425  g.  ? 

184.  126  g.  -  4%  of  126  g.  equal  how  many  Dg.  ?     dg.  ? 
eg.  ?     mg.  ? 

185.  A  nickel  weighs  5  g.     $5.00  in  nickels  weigh  how 
many  kilograms? 


164  DENOMINATE   NUMBERS 

186.  A  kilogram  is  equal  to  about  21  Ib.     Find  the  approxi- 
mate cost  of  a  kilogram  of  a  drug  that  costs  $  .60  a  pound. 

187.  What  is  the  cost  of  2242  g.  of  coffee  at  $  .60  a  kilogram  ? 

188.  If  a  kilogram  of  wool  costs  $  1.75,  how  much  will  6.5 
kilograms  cost  ? 

189.  The  denominations  of  time  measure  are  seconds  (sec.), 
minutes  (min.),  hours  (hr.),  days  (da.),  and  years  (yr.). 

TIME  MEASURE 

100         365         24        60        60 
century        yr.          da.         hr.       min.      sec. 

Till  out  the  following  table  of  equivalents : 
1  century  =  —  yr.  =  —  da.  =  —  hr.  =  —  min.  =  —  sec. 

190.  Find  the  ratio  of  one  hour  to  a  unit  of  each  denomina- 
tion of  time  measure. 

191.  Express  2  hr.  10  min.  as  sec.     As  min.     As  hr. 

192.  Express  f  yr.  as  da.     T5g-  da.  as  hr.     f  hr.  as  min. 

193.  Express  ^  of  a  year  as  da.     As  hr.     As  min.     As  sec. 

194.  Express  |f  min.  as  hr.     2|  hr.  as  da.     100  sec.  as  min. 

195.  Express  .12  yr.  as  da.     .33^  da.  as  hr.     .75  hr.  as  min. 
.66|  min.  as  sec.     .17  centuries  as  yr. 

196.  Express  108  sec.  as  a  decimal  of  a  minute.     Of  an  hour. 

197.  How  many  minutes  in  5%  of  an  hour?   In  65%?   95%? 

198.  1  hr.  +  15%  of  an  hour  =  how  many  minutes  ? 

199.  At  6%  now  much  interest  will  $300  gain  in  1  yr.  ?    In 
1|  yr.  ?    3  yr.  6  mo.  ?     2  yr.  9  mo.  ? 

200.  Express  in  years,  1  yr.  1  mo.  15  da.     Find  the  interest 
of  400  for  that  time  at  6  %  •     At  8  %  •    At  5  % . 


DENOMINATE   NUMBERS  165 

201.  Usually  every  fourth  year  has  366  da.,  and  is  called  a 
leap  year.     In  leap  year  the  month  of  February,  which  usually 
has  28  da.,  has  29  da.     How  many  hours  in  February  of  a  leap 
year  ? 

202.  Every  year  whose  number  is  divisible  by  4  is  a  leap 
year,  excepting  the  years  whose  number  ends  in  two  ciphers,  as 
1500,   1700,   1800.      Such  years  are  leap  years  only  if  their 
number  is  divisible  by  400,  as  1200, 1600,  2000.     Which  of  the 
following  are  leap  years  ? 

1848    1862    1892    1900    1904    2000    2108    2200    1000 

203.  At  a  dollar  a  day,  how  much  could  a  man  earn,  working 
6  da.  in  a  week,  in  the  month  of  February,  1896,  which  began 
on  Saturday  ?     How  much  in  February,  1898  ? 

204.  A£  $  1.50  per  day,  how  much  could  a  man   earn  in 
February  of  a  leap  year,  in  which  the  1st  of  February  fell  on 
Sunday  ? 

205.  A  man's  heart  beats  at  the  rate  of  about  72  beats  in  a 
minute.     At  that  rate  how  many  times  does  it  beat  in  an  hour  ? 
In  a  day  of  24  hr.  ?     In  a  common  year  ? 

206.  At  that  rate  how  many  times  would  a  man's  heart  beat 
in  a  lifetime  of  80  yr.,  -£  of  which  are  leap  years  ? 

207.  When  a  person  born  in  1883  is  20  yr.  old,  how  many 
years  of  his  life  have  been  leap  years  ? 

208.  The  time  in  which  the  earth  passes  once  around  the  sun 
is  365  da.  5  hr.  48  min.  46  sec.     That  is  how  much  more  than 
365  da.  ?     To  how  much  will  the  extra  time  amount  in  4  yr.  ? 
How  much  does  it  lack  of  being  a  whole  day  ? 

209.  In  adding  the  whole  day  to  every  fourth  year  or  leap 
year,  how  many  more  minutes  and  seconds  are  added  to  the 
year  than  rightly  belong  to  it  ? 


166  DENOMINATE   NUMBERS 

210.  Find  how  nearly  the  error  caused  in  a  century  by  this 
arrangement  is  corrected  by  omitting  the  extra  day  in  February 
at  the  end  of  the  century. 

211.  Find  the  amount  of  error  at  the  end  of  the  fourth  cen- 
tury, and  see  how  nearly  it  is  corrected  by  restoring  the  extra 
day  in  February  of  that  year. 

212.  Henry  rose  at  5.50  A.M.  and  went  to  bed  at  8.20  P.M. 
How  long  was  his  day  ? 

213.  CLASS   EXERCISE.     may  give  a  time  for  rising 

and  a  time  for  going  to  bed,  and  the  class  may  find  the  length 
of  the  included  day. 

214.  How  many  hours  were  there  in  the  year  1800  ? 

"  Thirty  days  hath  September, 
April,  June,  and  November. " 

215.  Learn  the  above  rhyme  and  remember  the  fact  that  all 
the  rest  of  the  months  except  February  have  31  days. 

216.  Write  the  names  of  the  months  in  order,  beginning  with 
January,  using  abbreviations.      Write  opposite  the  name  of 
each  month  the  number  of  days  it  contains. 

217.  Find  the  number  of  days  from  May  1st  to  June  7th. 

There  are  30  days  after  the  1st  in  May,  which,  with  the  7  days  in  June, 
make  37  days. 

Find  the  number  of  days  between  the  following  dates : 

218.  May  7,  1896,        July  4,  1896. 

219.  Jan.  1,  1900,         Mar.  1,  1900. 

220.  Sept.  28,  1899,   Nov.  5,  1899. 

221.  Dec.  15,  1899,   Jan.  31,  1900. 

222.  How  many  days  from  to-day  until  next  Christmas? 
Next  4th  of  July  ? 


DENOMINATE  NUMBERS  167 

' 


223.  CLASS  EXERCISE.     may  give  the  date  of  his  next 

birthday,  and  the  class  may  find  the  number  of  days  intervening. 

224.  Thirty  days  from  the  4th  of  July,  1876,  was  what  date  ? 

225.  What  was  the  date  60  days  after  Oct.  14th,  1492? 
Christmas,  1897  ? 

226.  On  the  17th  day  of  June,  Mr.  Herbert  borrowed  from 
a  bank  $  100  to  be  paid  in  60  days.     When  was  it  due  ? 

227.  What  will  be  the  date  30  days  after  to-day  ?     60  da.  ? 

228.  What  will  be  the   date  33  days  from  next  Monday  ? 
63  da.  ?     93  da.  ?     105  da.  ? 

229.  What  was  the  date  30  days  before  the  first  of  May, 
1891  ?    60  days  before  the  1st  of  March,  1892  ? 

230.  How  many  days  since  the  1st  of  January  of  this  year  ? 

231.  CLASS  EXERCISE.          -  may  give  the  date  of  his  last 
birthday,  and  the  class  may  find  how  many  days  have  passed 
since  then. 


Add 

da. 

hr. 

min. 

sec. 

da. 

hr. 

min. 

sec. 

232. 

17 

20 

30 

40 

233.  19 

19 

45 

30 

20 

16 

40 

10 

25 

20 

15 

30 

40 

18 

50 

20 

16 

12 

30 

30 

Add: 

da.  hr.  min.  sec.  da.  hr.  min.  sec. 

234.    19  14  30  45  235.    15  2  30  30 

70  20  45  15  30  10  45  25 

6  18  15  15  40  6  15  35 

4  2  30  30  9  8  15  15 


236.  Mr.  Cox  earns  $  2.00  for  each  day  of  10  hours  that  he 
works.  On  Monday  he  worked  8  hr.  30  min. ;  on  Tuesday,  9 
hr.  10  min. ;  Wednesday,  7  hr.  40  min. ;  Thursday,  8  hr.  30  min. ; 
Friday,  7  hr.  50  min ;  Saturday,  4  hr.  30  min.  How  much  did 
he  earn  in  that  week  ? 


168  DENOMINATE   NUMBERS 

237.  At  $  2.00  per  day  of  8  hours,  how  much  would  Mr.  Cox 
have  earned  ? 

Find  the  differences : 

da.       hr.     min.    sec.  da.  hr.  min.  sec. 

238.  47       18       2       10  239.    62  10  30  15 
25       20       1      40  48  20  19  45 


240.  Which  is  the  7th  month  ?     12th  ?     3d  ?     5th  ? 

241.  Which    month    is    February?      September?      June? 
November  ? 

242.  Bead  the  following  dates :    9/5/99.    10/5/98.    7/4/76. 

243.  Find  the  difference  of  time  between  March  5, 1898,  and 
Oct.  21,  1902. 

Write  as  below  and  subtract : 

1902     10    21 
1898      3      5 

In  finding  the  difference  between  two  dates  in  years,  months,  and  days, 
we  assume  that  30  days  =  a  month. 

American  Authors 

Ralph  Waldo  Emerson,  born  May  25,  1803  — died  April  27,  1882. 
John  G.  Whittier,  born  Dec.  17,  1807— died  April  27,  1892. 
Henry  W.  Longfellow,  born  Feb.  27,  1807  — died  March  24,  1882. 
James  Russell  Lowell,  born  Feb.  22,  1819  — died  Aug.  12,  1891. 

English  Authors 

Alfred  Tennyson,  born  Aug.  6,  1809  — died  Oct.  6,  1892. 
Charles  Dickens,  born  Feb.  7,  1812  — died  June  9,  1870. 

244.  Find  the  number  of  years,  months,  and  days  since  each 
of  the  authors  mentioned  above  was  born.     Since  each  died. 

245.  Find  the  age  of  each  author  at  his  death. 

246.  Find  the  time  between  the  birth  of  Emerson  and  that 
of  each  of  the  other  authors. 


DENOMINATE  NUMBERS  169 

247.  Find  the  time  between  the  death  of  Dickens  and  that 
of  each  of  the  other  authors. 

da.     hr.      min.     sec.  da.       hr.    min.  sec. 

248.  Multiply  7      18      20      10          249.    11      15      8      30 
by  7  9 

250.  Multiply  5  da.  10  hr.  20  min.  30  sec.  by  5.    By  6.    By  8. 

251.  If  you  spend  4  hr.  30  min.  in  school  every  day,  how 
many  hours  and  minutes  do  you  spend  in  a  school  week  ?     In 
a  school  month  of  4  wk.  ?     In  a  school  year  of  10  mo.  ? 

Divide : 

da.      hr.     min.    sec.  da.       hr.     min.    sec. 

252.  4)21      16      2      40  253.    6)13      13      13      30 

254.  Divide  15  da.  12  hr.  40  min.  30  sec.  by  5.    By  6.    By  8. 

255.  The  denominations  of  linear  measure  are  inches  (in.), 
feet  (ft.),  yards  (yd.),  rods  (rd.),  and  miles  (mi.). 

LINEAR  OR  LONG  MEASURE 

320         5|        3        12 
mi.    rd.         yd.       ft.       in. 

Fill  out  the  table  of  equivalent  values. 

1  mi.  =  — rd.  =  — yd.  =  — ft.  =  — in. 

256.  How  many  inches  in  2  yd.  1  ft.  7  in.  ?  5  yd.  3  ft.  7  in.  ? 

257.  Express  2  yd.  1  ft.  6  in.  as  in.     As  ft.     As  yd. 

258.  Express  2  mi.  20  rd.  as  mi.     As  rd.     As  yd.     As  ft. 

259.  1  ft.  is  what  part  of  a  yard  ?     Of  a  rod  ?     Of  a  mile  ? 
Reduce  complex  fractions  to  simple  fractions. 

260.  Express  1  ft.  6  in.  as  yd.     As  rd.     As  mi. 

261.  Express  1  yd.  1  ft.  9  in.  as  yd.     As  rd.     As  mi. 

262.  Express  T5^  mi.  as  rd.    T8T  rd.  as  yd.    %  yd.  as  ft.     -|  ft. 
as  in.          rd.  as  ft. 


170  DENOMINATE  NUMBERS 

263.  Express  f  in.  as  ft.    ^  ft.  as  yd.    |£  yd.  as  rd.    11$  rd. 
as  mi.     If  ft.  as  rd.     If  in.  as  ft.     4£  ft.  as  yd. 

264.  Express  .875  mi.  as  rd.     3.6  rd.  as  yd.     5.5  yd.  as  ft. 
.9  ft.  as  in.     1.66|  yd.  as  ft.     .64  rd.  as  yd. 

265.  Express  7.2  in.  as  a  decimal  of  a  foot.     As  a  decimal 
of  a  yard. 

266.  Express  115.5  ft.  in  yd.     In  rd.     In  mi. 

267.  How  many  rods  in  5%  of  a  mile  ?    In  15%  ?    35%  ? 

Add: 

yd.  ft.  in.                                                yd.  ft.  in. 

268.  2  1  Hi-  269.    6  2  7£ 
5  1  2j                                        3  2  llf 

mi.     rd.      yd.  ft.     in.  mi.       rd.    yd.  ft.     in. 

270.    6     200     2     1     10  271.    10       20     1     2     10 

8    120    1    2      6  7    300    1     1      8 


mi.     rd.      yd.  ft.     in.  mi.       rd.     yd.  ft.      in. 

272.    8     150     4    2     11  273.    10     180     4    2       3 

4     100     1     1       5  7      40    2     1     11 


mi.      rd.     yd.   ft.     in.  mi.      rd.     yd.   ft.      in. 

274.    16     200     3     2       4  275.    18         2     4     2     10 

14    150    1     1     10  5    319    2    1      7 


mi.       rd.     yd.    ft.    in. 
276.    15         3223 
10    319     1     1     9 

277.  Mr.  Smith's  lot  is  1  yd.  1  ft.  shorter  than  Mr.  Brown's 
lot,  which  is  30  yd.  long.     How  long  is  Mr.  Smith's  lot  ? 

278.  An  elm  tree  is  32  ft.  9  in.  high.     How  high  is  a  fir 
tree  that  is  6  ft.  10  in.  less  in  height  ? 


DENOMINATE   NUMBERS  171 

279.    Find  the  difference  between  the  height  of  John,  who 
is  5  ft.  3  in.  tall,  and  his  sister,  who  is  3  ft.  11  in. 


A        B  c       28°-      ^  ^  were  ^Of  miles  from  A  to 

C,  and  5J-  miles  from  A  to  J5,  how  far 
would  it  be  from  B  to  C  ? 

281.  If  it  were  20  mi.  20  rd.  from  A  to  C,  and  5  mi.  80  rd. 
from  A  to  J3,  how  far  would  it  be  from  BtoC? 

Multiply  : 

yd.    ft.    in.  mi.    rd.    yd.  ft.  in. 

282.  312  283.    52109 

3  4 


mi.     rd.    yd.    ft.  in.  mi.       rd.     yd.    ft.     in. 

284.    7     80    0    2     6  285.    10     160     4     1     11 

2  7 


286.  Multiply  2  mi.  240  rd.  3  yd.  5  ft.  6  in.  by  2.     By  3. 

287.  How  long  is  the  perimeter  of  a  regular  pentagon,  each 
of  whose  sides  is  3  yd.  1  ft.  9  in.  long  ? 

288.  A  summer  house  is  built  in  the  shape  of  a  regular 
hexagon,  each  side  being  2  yd.  1  ft.  6  in.     What  is  the  entire 
distance  around  it  ? 

289.  How  long  is  the  edge  of  the  border  of  a  flower  bed  in 
the  shape  of  an  octagon,  if  each  side  is  1  yd.  1  ft.  8  in.  long  ? 

290.  Divide  3  mi.  16  rd.  2  yd.  1  ft.  8  in.  by  2.     By  3.     By  4. 

291.  Divide  19  mi.  10  rd.  5  yd.  2  ft.  6  in.  by  5.     By  6.     By  7. 

292.  Divide  7  mi.  160  rd.  4  yd.  1  ft.  8  in.  by  2.   By  3.    By  4. 

293.  Henry  measured  a  cornstalk  and  found  it  to  be  5  ft. 
6  in.  long.     How  many  such  stalks  laid  in  a  continuous  line 
would  it  take  to  extend  a  mile  ? 


172  DENOMINATE   NUMBERS 

294.  Harry  has  two  dogs,  Don  Quixote  and  Sancho  Panza. 
Don  measures  4  ft.  3  in.  from  the  end  of  his  nose  to  the  tip  of 
his  tail,  weighs  77  Ib.  12  oz.,  and  is  3  yr.  7  mo.  9  da.  old. 
Sancho  measures  3  ft.  8  in.,  weighs  53  Ib.  14  oz.,  and  is  2  yr. 
11  mo.  28  da.  old.     Find  the  difference  of  the  lengths  of  the 
dogs.     Of  their  weights.     Of  their  ages. 

295.  Harry's  dog,  Sancho  Panza,  chased  a  rabbit  40  yd.  and 
then  gave  up  the  chase.    The  rabbit  had  10  yd.  the  start  of  the 
dog  and  ran  twice  as  fast  as  he  ran.     How  far  apart  were  the 
animals  when  Sancho  Panza  gave  up  the  chase  ?     Represent. 

296.  If  from  a  string  2  yd.  2  ft.  long,  2.5  ft.  is  broken  off  at 
one  end,  and  3.7  ft.  at  the  other,  how  long  a  string  is  left  ? 

297.  The  denominations  of  square  measure  are  square  inches 
(sq.  in.),  square  feet  (sq.  ft.),  square  yards  (sq.  yd.),  square 
rods  (sq.  rd.),  acres  (A.),  and  square  miles  (sq.  mi.). 

SURFACE  OR  SQUARE  MEASURE 

640  160  30£  9  144 

sq.  mi.  A.  sq.  rd.  sq.  yd.  sq.  ft.  sq.  in. 

Write  a  table  of  equivalent  values. 
1  sq.  mi.  =  —  A.  =  —  sq.  rd.  =  —  sq.  yd.  =  — sq.  ft.  =  —  sq.  in. 

SUGGESTION  TO  TEACHER.  Let  a  diagram  of  the  square  rod  be  drawn  on 
the  floor,  a  square  yard  in  one  corner  of  the  square  rod  being  subdivided 
into  square  feet,  and  one  of  the  square  feet  into  inches,  so  that  literally  each 
square  unit  may  be  a  part  of  the  units  of  higher  denominations.  Let 
pupils  make  many  practical  problems  upon  the  figures. 

298.  1  sq.  ft.  equals  what  part  of  a  sq.  yd.  ?    Of  a  sq.  rd.  ? 

299.  How  many  square  feet  in  2  sq.  rd.  ?     3  sq.  rd.  ?     5  sq. 
rd.  ?     7  sq.  rd.  ?     10  sq.  rd.  ?     12  sq.  rd.  ? 

300.  How  many  square  inches  in  5  sq.  ft.  60  sq.  in.  ?     In 
4  sq.  ft.  20  sq.  in.  ?     In  2  sq.  ft.  80  sq.  in.  ? 

301.  How  many  square  feet  in  4  sq.  yd.  6  sq.  ft.?     In  20 
sq.  yd.  7  sq.  ft.  ?     In  18  sq.  yd.  5  sq.  ft.  ? 


DENOMINATE   NUMBERS  173 

302.  How  many  square  yards  in  8  sq.  rd.  15  sq.  yd.  ?    In  10 
sq.  rd.  4f  sq.  yd.  ?     In  24  sq.  rd.  19  sq.  yd.  ? 

303.  Express  1  sq.  mi.  320  A.  80  sq.  rd.  in  sq.  rd.     In  A. 
In  sq.  mi. 

304.  Express  10  sq.  yd.  7  sq.  ft.  72  sq.  in.  in  sq.  in.     In  sq. 
ft.     In  sq.  yd. 

305.  Express  2  sq.  rd.  15  sq.  yd.  in  sq.  yd.     In  sq.  rd.     In 
sq.  ft.     In  A. 

306.  Express  ^  sq.  mi.  as  A.     -^  A.  as  sq.  rd. 

307.  Express  T8T  sq.  rd.  as  sq.  yd.    ^  sq.  yd.  as  sq.  ft.    -ff  sq. 
ft.  as  sq.  in.     -1/  sq.  in.  as  sq.  ft. 

308.  Express  .625  sq.  mi.  as  A.     .375  A.  as  sq.  rd.     .48  sq.  rd. 
as  sq.  yd.     .175  sq.  yd.  as  sq.  ft.     .7  sq.  ft.  as  sq.  in.     4.32  sq. 
in.  as  sq.  ft.     .18  sq.  ft.  as  sq.  yd. 

309.  Express  345.6  sq.  in.  as  a  decimal  of  a  square  foot. 

Add: 

sq.  yd.  sq.  ft.  sq.  in.  sq.  yd.  sq.  ft.  sq.  in. 

310.  21        2      100  311.    36        3       70 

786  15        7       60 

16       5        40  20        7       20 

Find  difference : 

sq.  yd.  sq.  ft.  sq.  in.  sq.  yd.  sq.  ft.  sq.  in 

312.    17        8        85  313.    21        6      100 
4        3       75  6       8        75 


Multiply : 

sq.  yd.  sq.  ft.  sq.  in.  sq.  yd.  sq.  ft.  sq.  in. 

314.    24       3      140  315.    16        4        96 
6  7 


316.    Multiply  2  A.  40  sq.  rd.  10  sq.  yd.  4  sq.  ft.  20  sq.  in. 
by  3.     By  4.     By  5. 


174  DENOMINATE   NUMBERS 

317.  How  many  acres  in  5%  of  a  square  mile  ?  7%  ?   15%  ? 

318.  1  sq.  ft.  —  37|%  of  a  square  foot  =  how  many  square 
inches  ? 

319.  Divide  by  3,  48  sq.  yd.  7  sq.  ft.  4  sq.  in. 

320.  Divide  by  5,  25  sq.  yd.  6  sq.  ft.  2  sq.  in. 

321.  Divide  20  A.  80  sq.  rd.  20  sq.  yd.  4  sq.  ft.  72  sq.  in.  by 
4.     By  8..    By  6. 

322.  How  many  square  feet  in  a  square  f  of  a  foot  in 
dimensions  ?     How  many  square  inches  ? 

323.  How  long  is  the  perimeter  of  a  square  f  of  a  foot  in 
dimensions  ?    What  fraction  of  a  square  foot  is  its  area  ?    How 
many  square  inches  in  its  area  ? 

324.  Mr.  Gilbert  owns  400  A.  120  sq.  rd.  of  land  in  Gibson 
County,  225  A.  and  10  sq.  rd.  in  Warrick  County,  and  14  A. 
40  sq.  rd.  in  Vanderburgh  County.     How  much  does  he  own  in 
those  counties  ? 

325.  A  farmer  had  80  A.  50  sq.  rd.  of  land.     After  selling 
30  A.  10  sq.  rd.,  how  much  had  he  left  ? 

326.  Mr.   Carter  owns   three   times  as  much   land  as  his 
cousin,  who  owns  120  A.  80  sq.  rd.     How  much  land  do  both 
own? 

327.  A  garden  180  ft.  long.  150  ft.  wide  is  surrounded  by  a 
tight  board  fence  6  ft.  high.     How  much  will  it  cost  to  paint 
the  fence  on  both  sides  @  12  ^  per  square  yard  ? 

SUGGESTION  TO  TEACHER.  Pupils  who  cannot  imagine  the  conditions 
of  this  problem  may  be  required  to  inclose  a  surface  on  their  desks  by 
a  strip  of  paper  folded  so  that  its  divisions  represent  the  parts  of  the 
fence. 

328.  From  each  corner  of  a  square,  a  side  of  which  is  2  ft. 
5  in.,  a  square  measuring  5  in.  on  a  side  is  cut  out.     Represent 
and  find  the  area  of  the  remainder  of  the  figure. 


DENOMINATE   NUMBERS  175 

329.  Find  the  area  of  the  walls  of  a  room  12  ft.  long,  10  ft, 
wide,  8  ft.  high. 

Find  the  areas  of  the  walls  and  ceiling  of  rooms  of  the 
following  dimensions,  and  the  cost  of  plastering  them  at  20 
cents  a  square  yard,  no  allowance  being  made  for  openings. 

a  b 

Length        Width         Height  Length       Width       Height 

ft.  ft.  ft.  ft.  ft.  ft. 

330.  20  18  10  40  30  12 

331.  30  25  9  21  20  9 

332.  25  21  9  30  18  8 

333.  18  15  8  4  20  9 

334.  15  12  8  16  15  8 

335.  Which  is  greater,  a  rectangle   12   in.  by  12  in.  or  a 
rectangle  16  in.  by  9  in.?   What  is  the  difference  in  the  length 
of  their  perimeters  ? 

336.  Give  dimensions  of  several  rectangles,  each  equal  to  a 
square  foot.     Compare  the  lengths  of  their  perimeters. 

337.  If  two  rectangles  have  equal  areas  but  different  shapes, 
which  will  have  the  longer  perimeter,  the  one  which  is  more 
nearly  square  or  the  other  ?     Illustrate. 

338.  Find  the  area  of  a  square  whose  perimeter  is  24  in. 
28  in.     36  in.     40  in. 

339.  Find  the  perimeter  of  a  square  whose  area  is  9  sq.  in. 

340.  The  denominations  of  cubic  measure  are  cubic  inches 
(cu.  in.),  cubic  feet  (cu.  ft.),  and  cubic  yards  (cu.  yd.). 

CUBIC  MEASURE 

27  1728 

cu.  yd.  cu.  ft.  cu.  in. 

Write  table  of  equivalent  values. 

1  cu.  yd.  =  —  cu.  ft.  =  — cu.  in. 


176  DENOMINATE   NUMBERS 

341.  How  many  cubic  inches  in  5  cu.  ft.  192  cu.  in.  ?     In 
1  cu.  yd.  624  cu.  in.  ? 

342.  How  many  cubic  inches  in  .875  cu.  ft.?     In  .625  cu.  ft.? 

343.  John  may  draw  a  square  yard  on  the  floor  in  a  corner 
of  the  room.     How  many  blocks  of  ice  1  foot  in  dimensions 
would  it  take  to  cover  that  square  yard  ? 

344.  If  another  layer  of  cubic  feet  of  ice  were  laid  upon  the 
first,  how  many  cubic  feet  of  ice  would  there  be  ?     How  high 
would  the  ice  be  piled  ? 

345.  If  a  third  layer  of  cubic  feet  of  ice  were  placed  upon 
the  other  two,  how  many  cubic  feet  of  ice  would  there  be  ? 

346.  What  name  is  given  to  a  solid  which  is  3  ft.  long,  3  ft. 
wide,  and  3  ft.  high  ? 

347.  In  the  square  yard  which  John  drew  William  may  set 
a  yard  stick  upright  at  that  corner  which  is  not  against  a  wall. 
Two  other  boys  may  place  sticks  in  such  a  position  that  a  cubic 
yard  is  outlined  in  the  corner  of  the  room. 

SUGGESTIONS  TO  TEACHER.  Devote  a  certain  space  in  the  room  to  the 
imaginary  cubic  yard.  See  that  every  member  of  the  class  images  a  cubic 
yard  in  that  particular  place.  Let  pupils  show  divisions  of  the  cube  by 
outlining  them  with  their  hands  in  the  space  devoted  to  it. 

Let  pupils  model  before  the  class  with  inch  cubes  the  figures  given  in 
the  following  exercises. 

348.  Image  a  cubic  yard  with  one  cubic  foot  cut  from  the 
upper  layer  at  a  corner  that  is  not  against  a  wall.     Model  the 
figure.     Outline  in  the  cubic  yard  in  the  corner  the  part  taken 
out.    What  is  the  ratio  of  the  part  taken  out  to  the  whole  cubic 
yard  ?    What  is  the  ratio  of  the  part  taken  out  to  the  part  left  ? 

349.  Take  two  more  cubic  feet  from  the  upper  layer,  one  on 
each  side  of  the  vacant  space.     Model.     Outline  in  the  cubic 
yard.     Tell  the  ratio  of  the  vacant  space  to  the  whole  cubic 
yard.     Of  the  vacant  space  to  the  filled  space. 


DENOMINATE   NUMBERS  177 

350.  Take  out  the  two  cubic  feet  that  were  directly  under 
the  cubic  foot  first  removed.     Model.     Outline.     Find  the  ratio 
of  the  vacant  space  to  the  cubic  yard.     Of  the  filled  space  to 
the  cubic  yard.     Of  the  vacant  space  to  the  filled  space. 

351.  Restore  the  whole  cubic  yard.     Take  away  the  middle 
cubic  foot  on  each  side  of  the  upper  layer.     Model.     Outline. 
Tell  the  ratio  of  the  vacant  space  to  the  cubic  yard.     Of  the 
filled  space  to  the  cubic  yard.    Of  the  vacant  space  to  the  filled 
space. 

352.  CLASS  EXERCISE;     may  give  directions  for  taking 

away  parts  of  the  cubic  yard.     The  class  may  tell  the  ratios 
of  the  spaces  to  one  another.     Some  members  may  model  the 
figures,  and  some  one  else  may  show  the  outline  of  the  space 
imaged  as  vacant  in  the  cubic  yard. 

353.  How  many  feet  in  the  sum  of  all  the  edges  of  a  cube  1 
yd.  in  dimensions  ? 

354.  How  much  will  a  cubic  yard  of  building  stone  cost  at 
$  2.50  a  cubic  foot  ? 

355.  How  many  cubic  feet  in  a  right  prism  6  ft.  long,  2  ft. 
wide,  and  1  ft.  high  ?     Model  the  prism. 

356.  How  many  cubic  feet  in  a  right  prism, 

a  4  ft.  long,  2  ft.  wide,  2  ft.  high  ? 
b  5  ft.  long,  3  ft.  wide,  1  ft.  high  ? 
c  8  ft.  long,  2  ft.  wide,  2  ft.  high  ? 
d  10  ft.  long,  4  ft.  wide,  2  ft.  high  ? 

357.  How  many  cubic  feet  in  a  tank  14  ft.  long,  10  ft.  wide; 
5  ft.  high  ? 

358.  How  many  cubic  yards  in  a  wall  81  ft.  long,  3  ft.  thick, 
and  9  ft.  high  ?    In  a  wall  30  ft.  long,  6  ft.  high,  and  3  ft.  thick  ? 

HORN.    GRAM.    SCH.    AR.  —  12 


178  DENOMINATE   NUMBERS 

359.  A  monument  is  in  the  shape  of  a  right  prism,  7  ft. 
long,  4  ft.  wide,  and  3  ft.  high.     How  much  will  it  cost  at 
$  3.50  per  cubic  foot  ? 

360.  If  the  engraving  on  it  costs  $  62.50,  what  will  be  the 
entire  cost  of  the  monument  ? 

361.  If  a  cake  3  in.  long,  3  in.  wide,  and  3  in.  high,  has 
icing  all  over  it  except  on  the  under  side,  how  many  square 
inches  of  icing  has  it  ? 

362.  To  cut  the  cake  into  inch  cubes,  how  many  cuts  would 
be  necessary  ?     How  many  inch  cubes  would  there  be  ? 

363.  How  many  of  the  cubes  would  have  icing  on  three 
sides  ?     On  two  sides  ?     On  one  side  ?     On  no  side  ? 

364.  How  many  inch  cubes  can  be  placed  on  a  square  foot  ? 
How  many  layers  of  those  cubes  would  it  take  to  make  a  cubic 
foot  ?     How  many  cubic  inches  in  a  cubic  foot  ?     What  is  the 
ratio  of  1  cu.  in.  to  a  cubic  foot  ? 

365.  Imagine  a  cubic  foot  of  marble  with  1  cu.  in.  cut  from 
each  of  the  upper  corners.     What  would  be  the  ratio  of  the 
part  cut  out  to  the  part  left  ? 

366.  Which  is  greater,  a  right  prism  12  in.  by  12  in.  by  12 
in.,  or  one  24  in.  by  12  in.  by  6  in.  ?     Compare  their  surfaces. 

367.  Give  dimensions  of  several  right  prisms,  each  of  which 
equals  a  cubic  foot.     Compare  their  surfaces. 

368.  How  many  cubic  feet  in  a  stick  of  timber  12  in.  wide, 
9  in.  thick,  and  24  ft.  long? 

369.  How  many  cubic  feet  in  a  cistern  5  ft.  square  and  6  ft. 
deep  ?     How  many  cubic  inches  ?     How  many  gallons  will  the 
cistern  hold  ?     (231  cu.  in.  =  1  gal.) 

Add: 

cu.  yd.     cu.  ft.         cu.  in.  cu.  yd.       cu.  ft.         cu.  in. 

370.  5    10     1700     371.   3     5    1400 
4     8     129         10    24     300 


DENOMINATE  NUMBERS                               179 
Add: 

cu.  yd.    cu.  ft.       cu.  in.  cu.  yd.    cu.  ft.      cu.  in. 

372.         9         11         1720  373.         4         15         1600 

6        18            10  8        20          200 


Subtract : 

cu.  yd.  cu.  ft.  cu.  in.                               cu.  yd.  cu.  ft.  cu.  in. 

374.     120  13  1700  375.       41  10  1634 

65  15  1125                            25  18  1507 


Subtract : 

cu.  yd.  cu.  ft.  cu.  in.                              cu.  yd.  cu.  ft.  cu.  in. 

376.       81  3           208  377.       16  4  800 

40  20          125                             4  21  525 


Multiply : 

cu.  yd.      cu.  ft.     cu.  in.  cu.  yd.      cu.  ft.      cu.  in. 

378.       20"         5         1160  379.       15         10           989 

3  4 


380.  Multiply  2  cu.  yd.  20  cu.  ft.  1000  cu.  in.  by  2.     By  4. 

Divide : 

cu.  yd.   cu.  ft.    cu.  in.  cu.  yd.  cu.  ft.  cu.  in. 

381.  5)    6         20         72  382.    8)    10        4        36 

383.  Divide  10  cu.  yd.  15  cu.  ft.  180  cu.  in.  by  3.     By  5. 

384.  Imagine  a  cubic  rod  of  marble.     Why  do  we  have  no 
such  measurement  as  a  cubic  mile  ? 

385.  A  pile  of  wood  8  ft.  long,  4  ft.  wide,  and  4  ft.  high  is  a 
cord  of  wood.     How  many  cubic  feet  in  a  cord  of  wood  ? 

386.  Bepresent  a  cord  of  wood  by  drawing  or  by  placing 
blocks  or  toothpicks. 

387.  How  many  cords  in  a  wood  pile  16  ft.  long,  8  ft.  wide, 
and  8  ft.  high? 


180  DENOMINATE  NUMBERS 

388.  At  $  5.00  per  cord,  what  is  the  value  of  a  pile  of  wood 
20  ft.  long,  4  ft.  wide,  and  4  ft.  high  ?     Of  a  pile  18  ft.  long, 
8  ft.  wide,  and  8  ft.  high  ? 

389.  The  standard  unit  of  metric  linear  measure  is  a  Meter, 
which  is  39.37  in.     This  length  was  obtained  by  calculating 
one  ten-millionth  of  the  distance  from  the  equator  to  a  pole  of 
the  earth.     A  kilometer  is  about  f  of  a  mile. 

METRIC  LINEAR  MEASURE 

1  kilometer  (Km.)     =  1000  meters 
1  hectometer  (Hrn.)  =  100  meters 
1  decameter  (Dm.)  =  10  meters 
1  meter  (m.) 

1  decimeter  (dm.)      =  .1  of  a  meter 
1  centimeter  (cm.)    =  .01  of  a  meter 
1  millimeter  (mm.)    =  .001  of  a  meter 

"Write  table  of  equivalents  : 
1  Km.  =  —  Hrn.  =  —  Dm.  =  —  m.  =  • —  dm.  =  —  cm.  = — mm. 

390.  Express  42  m.  as  centimeters.      As  decimeters.      As 
decameters.     As  hectometers. 

391.  Express  375  m.  as   kilometers.     As  decameters.     As 
decimeters.     As  millimeters. 

392.  Express  4287  m.  as  kilometers.     As  decimeters.     As 
hectometers. 

393.  How  many  centimeters  in  11%  of  12  m.  ?     Of  25  dm.  ? 

394.  7  %  of  192  m.  =  how  many  meters  ?     Decameters  ? 

SUGGESTION  TO  TEACHER.  Let  pupils  find  in  meters  and  decimals  the 
length  and  width  of  room.  Length  of  blackboards.  Length  of  diagonal 
of  room  or  blackboard.  Heights  of  pupils. 

395.  About  how  many  inches  in  a  decameter  ?     In  a  deci- 
meter ? 

396.  Find  approximately  the  number  of  inches  in  a  kilo- 
meter.    In  4  Hm.     In  12  Dm.     In  7  dm. 


DENOMINATE  NUMBERS  181 

397.  What  is  the  cost  of  12  m.  of  cloth  at  $.75  per  meter  ? 
Is   the   cloth  cheaper    or    dearer   than   at    $  .75    per  yard  ? 
Explain. 

398.  At  the  rate  of  36  Km.  per  hour,  how  far  will  a  train  run 
in  3  hr.  30  min.  ? 

399.  What  is  the  value  of  a  decameter  of  silk  at  $  1.65  per 
meter  ? 

400.  How  many  centimeters  long  is  the  perimeter  of  a  regu- 
lar octagon,  one  side  of  which  is  8  mm.  ? 

401.  Find  the  length  in  decimeters  of  one  side  of  a  regular 
pentagon  whose  perimeter  is  75  cm. 

402.  How  long  is  the  base  of  an  isosceles  triangle  whose 
perimeter  is  4  dm.  and  whose  equal  sides  are  each  12  cm.  ? 
Represent. 

403.  How  long  is  each  of  the  equal  sides  of  an  isosceles 
triangle  whose  perimeter  is  3  dm.  and  base  8  cm.     Construct. 

404.  A  kilometer  is  about  what  fraction  of  a  mile  ? 

405.  Find  approximately  the  number  of  miles  in  40  Km. 
In  72  Km.     In  3.2  Km.     In  6.72  Km. 

406.  Find  the  approximate  number  of  miles  in  9288  m. 

SUGGESTION.     Express  9288  m.  as  kilometers  before  finding  its  equiva- 
lent in  miles. 

407.  Find  approximately  the  number  of  miles  in  45864  m. 
In  63824  dm.     In  59888  Dm.     In  71848  Hm. 

408.  Find  approximately  the  number  of  kilometers  in  75  mi. 
In  235  mi.     In  84.5  mi. 

409.  How  many  square  millimeters  in  a  rectangle  1  cm.  long 
and  1  cm.  wide  ? 

410.  In  a  square  decimeter,  how  many  square  centimeters  ? 
Square  millimeters  ? 

411.  In    a    square   meter  how   many   square   decimeters? 
Square  centimeters  ?     Square  millimeters  ? 


182  DENOMINATE  NUMBERS 

412.  A  square  decameter  equals  how  many  square  meters  ? 
Square  decimeters  ?   Square  centimeters  ?  Square  millimeters  ? 

413.  A  square  hectometer  equals  how  many  square  deca- 
meters ?     Square  meters  ?     Square  decimeters  ? 

414.  A  square  kilometer  equals  how  many  square  hecto- 
meters ?     Square  decameters  ?     Square  meters  ?     Square  deci- 
meters ?     Square  centimeters  ?     Square  millimeters  ? 

415.  In  long  measure,  under  the  metric  system,  what  is  the 
ratio  of  a  unit  of  each  denomination  to  a  unit  of  the  next 
higher  denomination  ? 

416.  In  square  measure,  metric  system,  what  is  the  ratio  of 
a  unit  of  each   denomination  to  a  unit  of  the  next  higher 
denomination  ? 

417.  Write  a  table  of  square  measure,  metric  system.    Write 
a  table   of  equivalents   of   units   of  square  measure,  metric 

system. 

418.  Express    3  sq.  Km.  2  sq.  Dm.  50  sq.  m.   in    square 
meters.     In  square  decameters.     In  square  kilometers. 

419.  Express    7  sq.  m.   20   sq.  dm.   30   sq.  cm.  in  square 
millimeters.     In  square  centimeters.     In  square  decimeters. 

420.  Express  1  sq.  m.  2  sq.  dm.  3  sq.  cm.  in  sq.  mm.     In 
sq.  cm.     In  sq.  dm.     In  sq.  m.     In  sq.  Dm.     In  sq.  Hm.     In 
sq.  Km. 

421.  What  is  the  area  of  a  square  whose  perimeter  is  24 
cm.  ?    20  cm.  ?     40  mm.  ? 

422.  How  long  is  the  perimeter  of  a  square  whose  area  is  81 
sq.  cm.?     49  sq.  cm.?    64  sq.  dm.? 

423.  What  is  the   area  and  the  perimeter  of  a  rectangle 
which  is  35  cm.  long  and  ^  as  wide  as  long  ? 

424.  A  land  measurement,  10  meters  square,  or  its  equivalent, 
is  called  an  Are  (a.).     How  many  square  meters  in  an  are  ? 
How  long  is  the  perimeter  of  an  are  in  the  form  of  a  square  ? 


DENOMINATE  NUMBERS  183 

425.  What  is  the  cost  of  f  a.  of  land  at  $  12.50  per  are? 

426.  The  standard  unit  of  metric  land  measure  is  an  Are, 
which  is  equal  to  a  square  decameter  or  approximately  to  ^ 
of  an  acre. 

METRIC  LAND  MEASURE 

1  hectare  (Ha.)  =  100  ares 

1  are  (a.) 

1  centare(ca.)     =.01  are 

Notice  that  the  final  vowel  of  "hecto,"  and  "cento"  is  dropped  be- 
fore the  word  "  are." 

427.  How   many  meters   of  fence   would  be   required    to 
inclose  a  hectare  in  the  form  of  a  square  ? 

428.  Draw  on  the  floor  a  square  containing  a  centare.     How 
long  is  its  perimeter  ? 

429.  How  many  square  decimeters  in  a  centare?  In  an  are  ? 
In  a  hectare  ? 

430.  What  is  the  cost  of  24.7  Ha.  at  $425  a  Ha.?     Of 
63.25  Ha.  at  $1032  a  Ha.? 

431.  Approximately,  how  many  acres  in  280  a.  ?    In  160  a.  ? 
In  240  a.  ?     In  120  a.  ? 

432.  Find  the  approximate  value  in  ares  of  30  A.     75  A. 
17|  A.    8  A.  120  sq.  rd.    12  A.  80  sq.  rd.    6  A.    350  A.    500  A. 
40  sq.  rd. 

433.  Image   a   cubic  centimeter    and    a    cubic    decimeter. 
How  many  cubic  centimeters  are  equal  to  the  cubic  decimeter  ? 

434.  Draw  a  square  meter  in  one  corner  of  the  room.    Imag- 
ine it  covered  with  a  layer  of  cubic  decimeters  or  liters.     How 
many  are  there  ? 

435.  With  a  meter  stick  outline  a  cubic  meter.     How  many 
layers  of  cubic  decimeters  are  there  in  it  ?     How  many  cubic 
decimeters  ? 


184  DENOMINATE   NUMBERS 

436.  In  cubic  measure,  metric  system,  how  many  units  of 
each  denomination  make  one  unit  of  the  next  higher  denomina- 
tion ? 

437.  Write  the  table  of  cubic  measure,  metric  system. 

438.  A  cubic  meter,  or  its  equivalent,  is  called  a  Stere  (s.). 
Image  a  stere  of  ice,  of  cubical  form.     How  many  square  deci- 
meters in  all  its  surfaces  ? 

439.  Image  a  stere  of  marble,  2  m.  long  and  1  m.  wide. 
How  high   is  it?     How  many  decimeters  in   all  its  edges? 
Represent  with  blocks. 

440.  A  box  which  holds  a  stere  is  full  of  packages  of  Break- 
fast Food,  each  of  which  holds  a  liter.      How  many  packages 
are  there? 

441.  At  15^  a  liter,  what  is  the  value  of  the  contents  of  the 
box? 

442.  At   2^  a  liter,  what   is  the  cost  of  3  s.  of  wheat? 
4726  s.  ?    8347  s.  ? 

443.  The  standard  unit  of  metric  wood  measure  is  a  Stere, 
which  is  a  little  over  ^  of  a  cord. 

METRIC  WOOD  MEASURE 

1  decastere  (Ds.)  =  10  steres 

1  stere  (s.) 

1  decistere  (ds. )    =  .  1  of  a  stere 

444.  A  pile  of  wood  7  m.  long,  6  m.  wide,  and  5  m.  high 
contains  how  many  steres  ?     How  much  is  it  worth  at  $  1.50 
a  stere  ? 

445.  At  70^  a  stere,  what  is  the  value  of  a  pile  of  wood 
4.5  Dm.  long,  3.5  m.  wide,  and  300  cm.  high  ? 

446.  About  how  many  cords  are  there  in  20  s.  ?     In  32  s.  ? 
In  42  s.  ?     In  12.8  s.  ?     In  6.36  s.  ? 


DENOMINATE   NUMBERS  185 

447.  How  many  cubic  feet  in  a  cord?  About  how  many 
cubic  feet  in  a  stere  ? 

448 v  Approximately  how  many  steres  in  5  cd.  ?  In  7  cd.  ? 
In  9J  cd.  ?  In  3  cd.  64  cu.  ft.  ?  In  6  cd.  32  cu.  ft.  ?  In 
12  cd.  16  cu.  ft.  ?  In  4  cd.  8  cu.  ft.  ?  In  11  cd.  4  cu.  ft.  ?  In 
24  cd.  8  cu.  ft.  ? 

449.  Which  is  greater  and  how  much,  a  stere  or  a  kiloliter? 
Explain. 

450.  About  how  many  liquid  quarts  equal  a  liter? 
(See  page  159,  Ex.  115.) 

451.  About  how  many  quarts  are  there  in  a  decaliter?     In 
a  hectoliter  ?     In  a  kiloliter  ? 

452.  About  how  many  liters  in  7.7  qt.  ?     In  132  qt.  ?     In 
39  qt.  ?     In  17  qt.  ? 

453.  A  cask  of  oil  containing  187  1.  was  bought  at  20^  a 
liter  and  sold  at  25^  a  quart,  a  liter  being  counted  as  1.1  qt. 
How  much  was  gained  ? 

454.  How  much  is  gained  by  buying  209  1.  of  wine  at  30^ 
a  liter  and  selling  them,  at  40^  a  quart,  counting  a  liter  as 
1.1  qt.  ? 

455.  Image  a  milliliter.     What  else  is  it  called  ? 
(See  page  81.) 

456.  The  weight  of  a  cubic  centimeter  of  pure  water  at  its 
greatest  density  is  called  a  Gram  (g.). 

457.  Image  a  glass  vessel  of  cubical   shape   containing   a 
liter  of  pure  water.     How  many  grams  would  it  contain  ? 

458.  Approximately   1000  g.    equal   2|  Ib.      What   is   the 
approximate  equivalent  of  a  gram? 

MISCELLANEOUS  EXERCISES 

1.  Divide  62.5  by  .0025. 

2.  Find  the  g.  c.  d.  of  567  and  637. 


186  DENOMINATE  NUMBERS 


3.  Reduce  to  lowest  terms :     -J-JJ. 

4.  Find  the  1.  c.  m.  of  24  and  57. 

5.  Add  -fa  and  T4T. 

6.  From  T7-j  take  -fa- 

7.  Kesolve  into  prime  factors  26,460  and  60,060. 

8.  Write  the  improper  fraction  that  expresses  the  ratio  of 
the  first  prime  number  after  40  to  the  first  prime  number  after 
20,  and  reduce  it  to  a  mixed  number. 

9.  How  many  6ths  in  J  of  10  ?     In  J  of  7  ? 

10.  One  eighth  of  88  is  how  many  times  3  ?     4-J-  ?     5£  ? 

11.  If   13   is  a  divisor  and  39   a   dividend,  what  is   the 
quotient  ?     If  both  divisor  and  dividend  are  multiplied  by  4, 
what  is  the  quotient  ? 

12.  From  49J  subtract  a  number  which  is  ^  as  large. 

13.  A  minuend  is  15,  and  a  subtrahend  11.     What  is  the 
difference  ?     If  3  is  added  to  both  minuend  and  subtrahend, 
what  is  the  difference  ? 

14.  If  3^  is  added  to  both  minuend  and  subtrahend  in  the 
preceding  question,  what  is  the  difference  ? 

15.  Find  difference  between  %  of  ft  of  40  and  f  of  Jf  of  3. 

16.  Square  f     f.     f     ft. 

17.  What  fraction  multiplied  by  itself  will  give  f  ? 

18.  What  is  the  square  root  of  ff  ?    TW  ?    ft  ?    ft  ?    yinr ? 

19.  CLASS   EXERCISE.     may  give  a  fraction  that  is  a 

perfect  square,  and  the  class  may  give  its  square  root. 

20.  How  many  cubic  yards  of  earth  must  be  removed  to 
make  a  reservoir  120  ft.  long,  44  ft.  wide,  and  9  ft.  deep  ? 

21.  How  much  will  it  cost  to  dig  a  cellar  36  ft.  long,  18  ft. 
wide,  and  6  ft.  deep,  at  $  2.50  a  cubic  yard  ? 


MISCELLANEOUS  EXERCISES  187 


22.  How  many  cords  of  wood  in  a  pile  36  ft.  long,  4  ft.  wide, 
and  8  ft.  high  ?     At  $  3.50  a  cord,  how  much  would  it  cost  ? 

Find  the  cost  of  plastering  ceilings  of  the  following  rooms 
at  20  $  a  square  yard  : 

23.  18  ft.  x  20  ft.  27.  10  ft.  x  13|  ft. 

24.  16  ft.  x  17  ft.  28.  15£  ft.  x  18£  ft. 

25.  141  ft.  x  20  ft.  29.  27  ft.  x  36  ft. 

26.  9  ft.  x  16J-  ft.  30.  3  yd.  x  16  ft. 

31.  Estimate   the   cost   of   plastering  the  ceiling   of  your 
schoolroom  at  25^  a  square  yard. 

Find  the  cost  of  plastering  the  walls  and  ceiling  of  rooms  of 
the  following  dimensions  : 

length  width  height 

32.  6m.  5m.  2.8m. 

33.  8m.  7m.  3m. 

34.  6.5m.  5m.  3m. 

35.  7m.  6.5m.  3m. 

The  price  is  25  $  per  square  meter,  and  no  allowance  is  made 
for  openings. 

36.  An  arc  which  is  \  of  a  circumference  is  1  yd.  1  ft.  3  in. 
long.     How  long  is  the  circumference  ?     The  diameter  ?     The 
radius  ? 

37.  How  many  acres  in  a  field  56  rd.  long  and  40  rd.  wide  ? 

38.  How  much  will  it  cost  to  pave  a  walk,  60  ft.  long  and 
15  ft.  wide,  at  $  1.25  a  square  yard  ? 

39.  How  many  trees  can  be  planted  on  3  A.  of  ground  if 
only  1  tree  is  planted  on  each  square  rod  ? 

40.  How  many  cubic  feet  in  a  pile  of  wood  24  ft.  long,  3  ft. 
wide,  and  8  ft.  high  ?     How  many  cords  ? 


188  DENOMINATE   NUMBERS 

41.  Express  in  grams  the  weight  of  the  following  measure- 
ments of  pure  water  at  its  greatest  density : 

1  cu.  dm.      3  1.      15  cu.  cm.      1  ins.      1  ml.       Is.       4  cu.  m. 

42.  Express  the  measurements  above  in  kilograms. 

43.  Taking  2^  Ib.  as  the  equivalent  of  a  kilogram,  what  is 
your  weight  in  kilograms? 

44.  Express  the  following  in  avoirdupois  on  the  basis  of 
2£  Ib.  to  the  kilogram  : 

75  Kg.  88  Hg.  15  Dg.  175  g.  395  eg. 

45.  Express  the  following  in  avoirdupois  weight: 

33  Kg.  275  g.  924  mg.  99  Hg.  16  Dg. 

46.  How  much  is  gained  by  buying  a  barrel  of  flour  (196  Ib.) 
for  $  6.00  and  selling  it  at  7^  a  kilogram  ? 

47.  How  much  is  gained  by  buying  99  Ib.  of  sugar  at  5  f  a 
pound  and  selling  it  at  13  ^  a  kilogram  ? 

48.  How  much  is  gained  or  lost  by  buying  440  Ib.  of  dried 
fruit  at  10  ^  a  Ib.  and  selling  it  at  22  f  a  Kg.  ? 

49.  How  much  is  gained  by  buying  100  Kg.  of  coffee  at  50^ 
a  Kg.  and  selling  it  at  30^  a  Ib.  ? 

50.  How  much  is  gained  by  buying  500  Kg.  of  raisins  at 
12  j  a  Kg.  and  selling  them  at  8  ^  a  Ib.  ? 

51.  Imagine  a  cubic  decimeter  cut  from  each  corner  of  the 
upper  layer  of  a  cubic  meter,  and  find  the  surface  of  the  figure 
thus  formed. 

52.  A  sector  whose  arc  is  a  quadrant  was  cut  from  a  circle. 
If  the  area  of  the  whole  circle  was  4|  sq.  in.,  what  was  the 
area  of  the  part  that  was  left  ? 

53.  A  California  woman  took  300  Ib.  of  honey  from  her 
hives  in  a  month.     What  was  its  value  at  $  5.00  per  hundred- 
weight ? 


MISCELLANEOUS   EXERCISES  189 

54.  Earning  $  .75  per  day,  how  long  will  it  take  a  boy  to 
earn  enough  to  buy  a  $  12.00  watch  ? 

55.  Mr.  Taylor  bought  3  prize  pigs  whose  respective  weights 
were  3  cwt.  73  Ib.  12  oz.,  4  cwt.  99  Ib.  15  oz.,  5  cwt.  12  oz. 
How  much  did  they  all  weigh  ? 

56.  How  many  baskets,  each  holding  2J  pk.,  can  be  filled 
with  10  bu.  of  apples  ? 

57.  A  garden  containing  1089  sq.  yd.  is  49^-  yd.  long.     How 
wide  is  it  ? 

58.  A  fisherman  had  a  line  24  yd.  2  ft.  long.     A  fish  broke 
off  3  yd.  1  ft.  6  in.  of  it.     How  much  was  left  ? 

59.  A  dealer  bought  2  T.  3  cwt.  of  carpet  tacks  in  8-oz. 
papers.     How  many  papers  of  tacks  were  there  ? 

60.  How  long  is  one  side  of  an  equilateral  triangle  whose 
perimeter  is  5  yd.  1  ft.  3  in.  ?     Of  a  regular  pentagon  having 
an  equal  perimeter  ?     Of  a  regular  octagon  of  equal  perimeter  ? 

61.  A  string,  4  yd.  2  ft.  6  in.  long,  was  used  to  outline  a 
regular  hexagonal  flower  bed.     How  long  was  each  side  ? 

62.  A  farmer  sold  5  loads  of  hay,  each  Containing  17  cwt. 
85  Ib.     How  much  did  he  sell  ? 

63.  How  far  will  a  man  walk  who  begins  walking  at  9  A.M. 
and  walks  until  3.30  P.M.,  at  the  rate  of  5  mi.  an  hour  ? 

64.  A  family  started  to  go  in  a  wagon  to  St.  Louis  from  a 
town  132  miles  away.     They  rode  24  miles  a  day  for  5  days. 
On  the  morning  of  the  sixth  day,  they  started  at  9  o'clock  to 
ride  the  remaining  distance  at  the  rate  of  6  miles  an  hour.    At 
what  time  did  they  reach  St.  Louis  ? 

65.  Mr.  A  has  a  lot  40  rd.  square,  and  Mr.  B  has  a  lot  con- 
taining 40  sq.  rd.     How  many  more  square  rods  in  Mr.  A's  lot 
than  in  Mr.  B's  ? 


190  DENOMINATE   NUMBERS 

66.  A  flight  of  stairs  in  Mrs.  Long's  house  consists  of  18 
steps,  each  1  ft.  wide  and  8  in.  high.     How  much  will  the  stair 
carpet  cost  at  $  .75  per  yard,  if  3  in.  is  allowed  at  each  step 
for  the  turning  in  of  the  carpet  ? 

67.  How  much  can  be  earned  in  two  weeks  by  a  person  who 
earns  $  2.34  every  working  day  ? 

68.  General  McClellan  was  born  Dec.  3,  MDCCCXXVI,  and 
died  Oct.  29,  MDCCCLXXXV.      How  old  was  he  when  he 
died? 

69.  April  4th,  1898,  was  Monday.     At  the  close  of  that  day, 
Kuth  Mayo  found  that  there  were  8  weeks  and  3  days  left  of 
the  school  term.     On  what  day  did  the  term  close  ? 

70.  She  entered  college   Sept.  14,   1898.     The  first  term 
closed  Dec.  21.     How  long  was  it  ? 

71.  Her  expenses  for  the  term  were  $  95.75.     What  was  the 
average  per  week  ? 

72.  The  Thanksgiving  vacation  began  Nov.  24  and  ended 
Nov.  28,  and  there  was  no  other  vacation  in  the  term.     She 
attended  a  Saturday  class.      How  many  working   days   had 
she  in  that  term  ? 

73.  Her  second  term  began  Jan.  3,  1899,  and  ended  March 
25,  1899.     How  long  was  it  ? 

74.  Her  expenses  for  that  term  averaged  $  7.50  per  week, 
and  she  earned  $  25  during  the  term  by  outside  work.     Her 
expenses  were  how  much  more  than  her  earnings  ? 

75.  The  floor  of  Mrs.  Beed's  dining  room,  which  is  15  ft. 
long  and  14  ft.  wide,  is  laid  with  parquetry  flooring.     How 
much  did  it  cost  at  $  .621  per  square  yard  ? 

76.  The  wainscoting  is  3  ft.  high.     There  are  4  doors,  each 
3  ft.  wide.     Two  windows,  each  3  ft.  wide,  extend  down  into 
the  wainscoting  1 J  ft.    There  is  a  fireplace  4. ft.  wide.     How 
many  square  yards  in  the  wainscoting? 


MISCELLANEOUS  EXERCISES 


191 


77.  Her  dining  table  is  6  ft.  long  and  4  ft.  wide.     How  many 
square  yards  in  the  top  of  it  ? 

78.  A  rug  under  the  table  is  12  ft.  long  and  covers  12  sq.  yd. 
of  the  floor.     How  wide  is  it  ? 

79.  Make  a  problem  about  the  dimensions  of  a  room. 

80.  Fourteen  cords  of  wood  are  piled  evenly  on  an  open  car 
28  ft.  long  and  8  ft.  wide.     How  high  is  the  wood  piled  ? 

81.  If  a  leaf  of  a  book  is  12  cm.  long  and  9  cm.  wide,  how 
many  square  centimeters  in  the  surfaces  of  both  sides  of  the  leaf  ? 

82.  If  your  schoolroom  were  36  ft.  long  and  30  ft.  wide, 
how  many  square  yards  could  be  drawn  on  the  floor,  provided 

no  two  overlapped  ? 

83.  In  Pig.  1  the  angles  are  all  right 
angles.      How  long  is   the   line  repre- 
sented by  GH?     HA? 

84.  Copy  Fig.  1,  making  the  dimen- 
sions inches   or   centimeters.     Draw   a 

H  G     construction  line  BE.     How  long  is  it  ? 

Find  the  area  of  Fig.  1  by  finding  the 
sum  of  the  areas  of  the  two  rectangles 
that  are  thus  formed. 

85.  Copy  Fig.  2.     Find  its  area  by 
drawing  a  construction  line  from  C  per- 
pendicular to  FE  and  finding  the  area 
of    the   two  rectangles   that  are  thus 

FIG.  2.  formed. 

86.    Find  the  area  of  Fig.  2  by  drawing  a  construction  line 
A  BE      F      from  O  perpendicular  to  AF  and  finding 

the  area  of  two  rectangles  thus  made. 

87.    Copy  Fig.  3,  making  AB  6  in., 
EC  3  in.,    CD  3  in.,    DE   3  in.,   EF 
3  in.,   FG  8  in.     Find  length  of  GH 
FIG.  3.  and  HA. 


FIG.  1. 


10 


E 


192 


DENOMINATE   NUMBERS 


88.  Find  area  of  Fig.  3  by  drawing  construction  lines  that 
will  divide  it  into  three  rectangles  and  finding  the  area  of  the 
rectangles.     Show  different  ways  of  dividing  it. 

89.  Find   area  of  Fig.   3    by   finding  area  of   a  rectangle 
AFQH  and  subtracting  the  square  BE  DC. 

90.  Copy  Fig.  4,   making  AB  5  in., 
BC  3  in.,  CD  4  in.,  DE  6  in.,  EF  2  in., 
FG  4  in.     GH=?     HA=  ? 

91.  Show  four  different  ways  of  di- 
viding   Fig.    4    into    three   rectangles. 
Find  its  area. 

jT~  a         92.    Beginning  at  a  point  marked  A, 

FIG.  4.  £raw  to  the  rig]^  4  in<?  down  3^  to  tne 

right  3,  down  4,  to  the  left  3,  down  3,  to  the  left  4,  up  3,  left  3, 
up  4,  right  3,  up  to  A.  Find  perimeter  of  the  figure.  Find 
its  area. 

93.  Beginning  at  A,  draw  down  7,  to  the  right  3,  up  3,  right 
3,  down  3,  right  4,  up  4,  left  2,  up  2,  left  3,  up  1,  left  5.  Show 
several  different  ways  of  dividing  the  figure  into  rectangles. 
Find  its  area. 


94.    CLASS  EXERCISE. 


may  give  directions  to  the  class 


for  drawing  a  figure  which  has  only  straight  lines  and  right 
angles.  The  class  may  divide  the  figure  by  different  construc- 
tion lines  and  find  their  length  and  the  area  of  the  figure. 

95.  ABCD  is  a  square  9  in.  in  dimen- 
sions, and  EFGH  is   a   square  5  in.  in 
dimensions.     How   many   square   inches 
in  the  surface  lying  between  the  perim- 
eter of  the  squares  ? 

96.  The  frame  of  a  mirror  is  28  in. 
long  and  20  in.  wide  on  the  outside  edge. 
The  glass  in  the  center  of  the  frame  is 

Eepresent.     What  is  the  width 


FIG.  5. 
20  in.  long  and  12  in.  wide. 


MISCELLANEOUS   EXERCISES 


193 


of  each  side  of  the  frame  ? 
surface  of  the  frame  ? 


How  many  square  inches  in  the 


97.  A  rug  12  ft.  long  and  9  ft.  wide  was  laid  on  the  floor, 
leaving  a  margin  3  ft.  in  width  all  around  the  rug.     What  was 
the  area  of  the  floor  ?     Of  the  rug  ?     Of  the  uncovered  part  ? 

98.  A  picture  18  in.  long  and  15  in.  wide  has  a  frame  each 
side  of  which  is  6  in.  wide.     How  many  square  inches  in  the 
surface  of  the  frame  ? 

99.  A  door  7  ft.  high  and  3  ft.  wide  has  a  6-inch  casing 
around  it.      How  many  square  inches  in  the  surface  of  the 
casing  ? 

100.    Measure  a  door  and  the  width  of  its  casing  and  find  the 
number  of  square  feet  in  the  surface  of  the  casing. 

101.  ABCD  represents  a  square  15  in. 
in  dimensions.    The  altitude  and  the  base 
of  each  triangle  is  4  in.     Eind  the  area 
of  the   octagonal   figure   left   when   the 
triangles  are  cut  away. 

102.  Two  lines  are  respectively  6  in. 
and  10  in.    What  is  their  average  length  ? 

103.  Reproduce  the  trapezoid  ABCD, 
making  AB  4  in.,  BC  8  in.,  and  AD  6  in. 
Let  x  be  the  middle  point  of  AB,  and 
y  the  middle  point  of  DC.     xy  repre- 
sents the  average  length  of  the  parallel 

FIG.  7.  sides  of  the  trapezoid.     How  long  is  xy  ? 

104.  Through  the  point  y  draw  the  line  EF  parallel  to  AB. 
EF  is  the  altitude  of  the  trapezoid. 
Find  the  area  of  the  rectangle  AEFB. 
Cut  off  the  triangle  yFC  and  apply  it 
to  the  triangle  yED.  How  does  the 
^  area  of  the  rectangle  AEFB  compare 
with  the  area  of  the  trapezoid  ADCB  ? 


FIG.  8. 


HORN.    GRAM.    SCH.    AR.  — 13 


194  DENOMINATE   NUMBERS 

105.  Draw  another  figure  and  show  the  reasons  for  the  fol- 
lowing rule. 

To  find  the  area  of  a  trapezoid  — 

Multiply  the  average  length  of  the  parallel  sides  by  the  altitude. 

106.  Draw  a  trapezoid  whose  parallel  sides  are  respectively 
9  in.  and  5  in.,  and  whose  altitude  is  6  in.     Find  its  area. 

107.  A  board  is  16  ft.  long,  2  ft.  wide  at  one  end  and  1  ft. 
wide  at  the  other,  tapering  gradually.     How  many  square  feet 
in  the  surface  of  the  board  ? 

108.  A  farmer  has  a  field  in  the  shape  of  a  trapezoid.     One 
of  the  parallel  sides  is  40  rd.  long,  the  other  is  24  rd.  long. 
The  distance  between  them  is  25  rd.     Find  area  of  the  field. 

Make  diagrams  to  illustrate  the  following  problems  : 

109.  Mrs.  Hall's  parlor  has  a  bay  window,  the  floor  of  which 
is  in  the  shape  of  a  trapezoid.    The  longer  of  the  parallel  sides 
is  12  ft.,  the  shorter  9  ft.,  and  the  distance  between  them  is 
4  ft.     How  much  will  it  cost  to  cover  it  with  parquetry  floor- 
ing at  $  1.25  per  square  yard  ? 

110.  Her  sitting  room  is  18  ft.  long  and  15  ft.  wide.     How 
many  strips  of  carpet  1  yd.  wide,  running  lengthwise  of  the 
room,  will  be  required  to  cover  the  floor  ?    How  many  yards  in 
each  strip  ?     How  much  will  the  carpet  cost  at  $  ,87-J-  a  yard  ? 

111.  How  many  strips  of  carpet  1  yd.  wide,  running  length- 
wise, will  be  required  for  a  floor  24  ft.  wide  ?     If  the  floor  is 
30  ft.  long,  how  many  yards  will  be  required  ? 

112.  How  much  will  it  cost  to  carpet  a  room  30  ft.  long  24 
ft.  wide  with  ingrain  carpet  at  $  .75  per  yard,  if  a  margin  of 
^  yard  is  left  uncovered  ? 

113.  Mrs.  Eoss  covered  the  floor  of  her  parlor,  20  ft.  long 
and  18  ft.  wide,  with  velvet  carpet  27  in.  wide.     How  many 
strips  were  used  ?     What  was  the  cost  of  the  carpet  at  $  1.65 
per  yard  ? 


MISCELLANEOUS  EXERCISES  195 

114.  Her  sitting  room,  16  ft.  by  12  ft.,  is  carpeted  with  in- 
grain carpet  at  $  .67  per  yard.     In  order  to  match  the  figures, 
the  carpet  layer  was  obliged  to  cut  off  or  waste  a  piece  of  car- 
pet ^  of  a  yard  long  from  each  strip  except  the  first.     How 
much  did  the  carpet  cost? 

115.  Allowing  ^  of  a  yard  to  be  turned  in  or  cut  off  from 
each  strip  except  the  first  in  order  to  match  the  figures,  how 
much  will  it  cost  to  carpet  a  room  21  ft.  by  18  ft.  with  carpet 
27  in.  wide,  worth  $  1.35  per  yard  ? 

SUGGESTION   FOR   CLASS   EXERCISE.     Let  pupils  give   dimensions   of 
floors  and  estimate  cost  of  covering  them  with  carpet  of  different  widths. 

116.  What  is  the  volume  of  an  8-inch  cube  ? 

117.  What  number  cubed  equals  27  ?     216.? 

118.  Find  value  of  x  :  or5  =  64.     y?  =  125.     0^  =  343. 

119.  Image  a  box  in  the  shape  of  a  cube  whose  volume  is 
8  cu.  in.     Suppose  the  box  to  be  entirely  covered  with  blue 
velvet.     How  many  square  inches  of  the  velvet  are  there  ? 

120.  When  a  3-inch  cube  is  built  of  inch  cubes,  how  many 
inch  cubes  are  there  that  cannot  be  seen  from  the  outside  in 
whatever  position  the  cube  may  be  placed  ? 

121.  How  many  of  the  cubic  centimeters  that  make  a  cubic 
decimeter  have  any  of  their  surfaces  on  the  outside  of  the 
figure  ?    How  many  have  one  surface  ?    Two  surfaces  ?    Three 
surfaces  ?     No  surface  ? 

122.  If  a  pipe  discharges  245  gal.  2  qt.  1  pt.  in  1  hr.,  how 
much  will  it  discharge  in  the  time  from  Tuesday,  6  P.M.,  to 
Wednesday,  11  A.M.? 

123.  A  school  that  uses  12  crayons  in  a  week  will  use  how 
many  gross  of  crayons  in  40  weeks  ? 

124.  A  stationer  bought  8  gross  of  lead  pencils  and  sold 
50  dozen  of  them.     How  many  lead  pencils  had  he  left? 


196  DENOMINATE   NUMBERS 

125.  If  he  bought  them  at  $  3  per  gross  and  sold  them  at 
5  $  apiece,  how  much  did  he  gain  on  each  pencil  ?     On  50  doz. 
pencils  ? 

126.  At  10^  a  square  yard,  how  much  will  it  cost  to  sod  a 
lawn  40  ft.  long  and  36  ft.  wide  ? 

127.  If  you  had  $23.70  in  the  bank  at  4%  interest,  how 
much  interest  would  it  yield  you  each  year  ? 

128.  Mr.  Thomas  Kepler  bought  of  Mr.  Prank  Barton  an 
overcoat  and  a  vest,  the  price  of  which  was  $  27.     Mr.  Kepler 
gave  in  payment  a  check  for  that  amount  on  the  First  National 
Bank  of  Denver,  Colorado,  of  which  the  following  is  a  copy : 


SUGGESTIONS  TO  TEACHER.  Explain  the  method  of  using  checks  to  make 
payments.  Get  a  check  book  from  a  bank.  Select  some  pupil  to  act  as 
banker  and  let  pupils  make  imaginary  purchases  from  one  another,  giving 
checks  for  the  necessary  amounts. 

129.  What  is  the  advantage  of  keeping  money  in  a  bank  and 
drawing  it  out  as  it  is  needed  ? 

130.  Mr.  Dow  borrowed  $  800  at  5%  from  Mr.  Howe,  kept 
it  3  yr.,  and  then  gave  Mr.  Howe  his  check  for  the  amount 
due.     Make  out  the  check. 

131.  Mr.  Ford  had  $  427  in  bank.     He  drew  out  $  135.87, 
deposited  $  77.50,  then  drew  out  $  35.25.    How  much  remained 
to  his  credit  in  the  bank  ? 


MISCELLANEOUS   EXERCISES  197 

132.  Mr.  Arnold  had  $  1200  in  the  bank.     On  Monday  he 
drew  out  $  60.     On  Tuesday  he  drew  out  $  30  more  than  on 
Monday.      On  Wednesday  he  drew  out  $  90  more  than  on 
Tuesday.     How  much  had  he  left  in  the  bank  ? 

133.  Mr.  Monroe  lent  $  100  at  5%  interest.     At  the  end  of 
7  yr.  the  principal  (that  is  the  sum  lent)  and  the  interest  were 
both  paid.     To  how  much  did  they  both  amount  ? 

134.  What    amount    will    Mr.    Day    receive    from     $  228 
which  he  lent  2  yr.  ago  at  6%,  if  both  principal  and  interest 
are   paid  ?      How   much   if    the    rate    of    interest   is 


135.  CLASS  EXERCISE.     -  niay  name  a  sum  of  money 
supposed  to  be  lent  at  3|-%  for  2  yr.,  and  the  class  may  find 
the  interest  and  the  amount  of  principal  and  interest. 

136.  Mr.  Shaw  borrowed  $  750,  kept  it  until  it  had  gained 
$  78.75  interest,  and  then  paid  $300.      How  much  did  he 
still  owe  ? 

137.  Mr.  Shaw  borrowed  $  600  and  gave  his  note  for  it,  due 
in  2  yr.  with  6%  interest.     How  much  was  due  at  the  end  of 
the  two  years  ?     At  that  time  he  made  a  partial  payment  of 
the  note,  paying  only  $  200.     How  much  did  he  still  owe  ? 

138.  Mr.  Shaw  borrowed  '$  700,  giving  his  note  at  5%  in- 
terest.    At  the  end  of  2  years  how  large  a  partial  payment 
must  he  make  that  only  $  500  may  be  due  ? 

139.  Mr.  Shaw  borrowed  f  900  at  6%.     At  the  end  of  the 
first  year  he  paid  $  154.     How  much  was  still  due  ?     That 
sum  went  on  gaining  interest  until  the  end  of  the  second  year  ; 
then  he  paid  f  148.     At  the  end  of  the  third  year  he  paid  all 
that  was  due.     How  much  did  he  pay  ? 

140.  CLASS  EXEKCISE.     -  may  name  a  sum  of  money  and 
a  number  of  years  for  which  it  was  borrowed.     Other  members 
of  the  class  may  suggest  partial  payments  to  be  made  at  dif- 
ferent times,  and  the  class  may  find  the  amount  due  after  each 
payment. 


198  DENOMINATE  NUMBERS 

141.  Make  out  a  bill  for  the  following  goods  and  receipt  it: 
C.  H.  Wilson  bought  from  J.  G.  Cooper  &  Co.,  at  Columbus, 

Ohio,  on  the  tenth  day  of  June,  1875:  13  Ib.  coffee  @  30^; 
4  Ib.  butter  @  35  f ;  10  Ib.  flour  @  6  ^ ;  12  Ib.  dried  beef  @  24  f ; 
25  Ib.  sugar  @  18  ^j  3  Ib.  starch  @  20 /. 

142.  From  a  field  containing  400  sq.  rd.,  the  owner  sold  a 
piece  of  land  15  rd.  square  and  another  piece  containing  15 
sq.  rd.     How  many  square  rods  had  he  left  ? 

143.  Mr.  Eay  sold  his  house  and  a  farm  of  75  A.,  receiving 
$  7500  for  both.     If  the  house  was  worth  $  2000,  how  much 
did  he  receive  per  acre  for  the  land  ? 

144.  If  Mr.  Eudd  earns  $  15  a  week  and  spends  $  7,  in  how 
many  weeks  will  he  earn  $  100  ? 

145.  Mrs.  Hall's  sitting  room  has  a  picture  rail  extending  all 
around  it  1^  ft.  from  the  ceiling.     The  room  is  18  ft.  long  and 
15  ft.  wide.     How  much  did  the  picture  rail  cost  at  7 J  ^  per 
foot? 


CHAPTER  VI 

ALIQUOT    PARTS 

1.  Numbers,  either  integral  or  fractional,  by  which  a  given 
number  is  divisible  are  called  Aliquot  Parts  of  that  number. 
For  example,  5  and  21  are  aliquot  parts  of  10. 

Give  three  numbers  which  are  aliquot  parts  of  100. 

2.  Draw  three  vertical  lines  each  10  in.  long 
and  divide  them  into  lengths  each  2J  in.   long. 
Number    the   lengths    consecutively   as   in   the 
diagram.     How  many  2J  in.  lengths  in  10  in.  ? 
In  20  in.  ?     In  30  in.  ? 

3.  Beginning  with  21-,  count  quickly  by  inter- 

10-1   20-J   30-J    vals  of  2i  to  30-     Count  back  from  30  to  2£  by 
intervals  of  2^. 

NOTE  TO  TEACHER.  The  following  exercises  are  for  rapid  drill,  which 
should  be  given  frequently  until  pupils  learn  the  ratios  of  the  smaller  ag- 
gregations of  2^  to  one  another.  This  kind  of  work  leads  to  expertness 
in  business  calculations. 

4.  How  many  times  2|  is  7£?  17|?  27£?  12 J?  25? 
15?  22£?  10?  20?  30? 

6.  Give  quickly  the  4th  multiple  of  2J.  The  7th.  10th. 
5th.  3d.  6th.  9th.  llth.  8th.  12th.  2d. 

6.  Which  multiple  of  2J  is  7|?  15?  25?  10?  17£? 
224?  30?  121?  "274?  20?  5? 


7^  How  many  times  2£  must  be  added  to  7-J-  to  make  20  ? 

30?  121?     17£?     27J?     221?     15?     25?     10? 

8.  How  many  times  2^  must  be  taken  from  27^-  to  leave 

15?  7£?     22^?     12^?    2£?     10?     5?     17|?     25?    20? 


199 


200 


ALIQUOT  PARTS 


9.  With  15  as  a  starting  point  find  how  many  times  2J 
must  be  added  to  it  or  subtracted  from  it  to  equal  27£.  12£. 
2  30.  20.  7  22.  10.  17  5.  25. 


10.  Learn  to  give  quickly  the  ratio  of  5  to  each  of  the  mul- 
tiples of  2J  that  are  less  than  32^. 

11.  Take  Ex.  10,  substituting  for  5  each  multiple  of  2J  that 
is  greater  than  5  and  less  than  32^-. 

12.  CLASS  EXERCISE.     -  may  name  two  multiples  of  2^-, 
and  the  class  may  give  the  ratio  of  the  greater  to  the  less,  then 
the  ratio  of  the  less  to  the  greater. 


Cancel  : 
10  x 


' 


x  6 


7fxl5 


=  9 


7x2ixl2i 


16 

' 


25  x  15  x  3 
12^x22^x6  = 


25  x  9  x  21  x  7 

17.    At  21  $.  per  yard  what  is  the  cost  of  8  yd.  of  lace  ?     12 
yd.  ?     7  yd.  ?     9  yd.  ?     6  yd.  ?     11  yd.  ? 

18.    At  21^  per  yard,  how  many  yards  of 
lace  can  be  bought  for  10  f?     20  f  ?     30^  ? 

50^?     121^?     25^? 


36%- 


10J 


40- 


19.  Draw  four  vertical  lines  each  10  in. 
long  and  divide  them  into  lengths  of  3^-  in. 
Number  the  lengths  consecutively.  40 
in.  equals  how  many  times  3J  in.  ? 

to  40  by  intervals  of  3£. 


similar  to  those  in  Exs. 


20.  Learn  to  count  quickly  from 
From  40  to  0  by  intervals  of  31 

NOTE  TO  TEACHER.     Give  exercises  upon 
4-12  upon  2|. 

21.  At  3-^  per  yard,  what  is  the  cost  of  7  yd.  of  lace  ? 
10  yd.  ?     8  yd.  ? 


ALIQUOT  PARTS  201 

22.  At  3J^  per  yard,  how  many  yards  of  calico  can  be  bought 
for  20^?     40^?     10^?     30^? 

23.  How  many  more  yards  of  ribbon  can  be  bought  for  40  f 
when  the  price  is  2J  ^,  than  when  it  is  3^  $  ? 

Cancel : 

31x7^x5     =9  21x61x121-^ 

'   6fx3x2ix7  10x3-ix9 

12ix6xlO=9  13|x2jx2=9 

'25x7^x11  6fx7ix5 

28.  A  merchant  bought  goods  at  8  ^  a  yard,  and  sold  them 
for  10 1  a  yard.     How  much  did  he  gain  on  each  yard  ?     What 
is  the  ratio  of  the  gain  to  the  cost  ? 

29.  Find  ratio  of  gain  to  cost  of  goods : 

a      b        c          d          e  f          g  li 

Bought  at  6^      8^      9?      5/        1\j      10  X      10^ 
Sold  at        9^    12^    12^      7J^    10 f        12J^    15^ 

30.  When  goods  are  bought  for  9 ^  a  yard  and  sold  for  6^  a 
yard,  how  much  is  lost  ?     What  is  the  ratio  of  the  loss  to  the 
cost? 

31.  Find  ratio  of  the  loss  to  the  cost  of  goods: 

a        b         c         d         e         f          g  7i 

Bought  at     8^    10^    15^    20^    18^    1\f    10^ 
Sold  at  6^      8^    10 #     15^     15 £    5^        7|^ 

32.  Find  ratio  of  gain  or  loss  to  cost  of  goods : 

a          b         c          d         e         f         g          h 
Bought  at       45^    20^    25^    36^    22^    33^    36^    60^ 
Sold  at  50^    15^    30^    40^    33^    22^    32^ 

33.  Find  ratio  of  gain  to  cost  of  goods  : 

a          b  c          d  e          f 

Cost,  10^      20^      30^        6f^    16f^    26|^ 

Selling  price,  13^    23$  f    33^^    10^      20^      30^ 


202  ALIQUOT  PARTS 

34.  How  much  is  gained  on  each  apple  bought  at  the  rate 
of  2  for  5  ^,  and  sold  at  3  f  apiece  ? 

35.  On  the  4th  of  July,  Andrew  bought  lemons  at  the  rate 
of  3  for  a  dime.     He  used  one  lemon  and  f  of  a  cent's  worth 
of  sugar  to  make  each  glass  of  lemonade.     How  much  did  he 
gain  on  each   glass  of  lemonade  sold  at  10^  a  glass?     How 
much  on  l^  doz.  glasses  ? 

36.  For  $  1.00  William  bought  ice  cream  enough  to  fill  36 
ice  cream  plates.     He  sold  it  at  10  f  a  plateful.     How  much 
did  he  gain  on  each  plateful  of  ice  cream  ?     He  sold  1^-  doz. 
platefuls.     How  much  did  he  gain  ? 

37.  The  rest  of  his  ice  cream  was  unsold  and  was  wasted. 
Did  he   gain   or   lose,   and   how   much,  on  his  whole  trans- 
action ? 

38.  Beginning  at  16f,  count  quickly  to  100  and  back  to  0 
by  intervals  of  16|. 

39.  How  many  times   is  16|  contained  in  66J  ?     In  33£  ? 
In  100?     In83J? 

40.  How  many  times  16J  must  be  added  to  50  or  subtracted 
from  it  to  make  831  ?     100  ?     66|  ?     16|  ? 

41.  Give  the  ratio  of  16|  to  each  of  its  multiples  that  is 
less  than  101. 

42.  Learn  to  give  quickly  the  ratio  of  each  multiple  of  16| 
that  does  not  exceed  100,  to  every  other  multiple  of  16|  that 
does  not  exceed  100. 

43.  CLASS  EXERCISE.     may  name  two  multiples  of  16  j, 

and  the  class  may  give  the  ratio  of  the  less  to  the  greater,  and 
of  the  greater  to  the  less. 

44.  At  16f  $  per  yard,  what  is  the  cost  of  3  yd.  of  lawn  ?     5 
yd.  ?     7  yd.  ?     2  yd.  ?     6  yd.  ?     4  yd.  ? 

45.  At  16 \$  per  yard,  how  many  yards  of  lawn  can  be 
bought  for  $1?    $2?    $7?    $10?    33^?    83^? 


ALIQUOT  PAKTS  203 

46.  Beginning  with  8^,  count  quickly  to  100  and  back  to  0 
by  8Vs. 

SUGGESTION  TO  TEACHER.     Exercises  should  be  given  upon  8|  similar 
to  those  in  Exs.  39-43  upon  16f  . 

47.  CLASS  EXERCISE.        —  may  name  two  multiples  of  8J, 
and  the  class  may  give  their  reciprocal  ratios. 

48.  8^  =  25%  of  what  ?     50%  ?     10%  ? 

49.  At  8J  f  per  yard,  what  is  the  cost  of  6  yd.  of  muslin  ? 
3  yd.  ?     12  yd.  ?    4  yd.  ?     8  yd.  ?     10  yd.  ?     5  yd.  ?     9  yd.  ? 

50.  At  8  J  f  per  yard,  how  many  yards  of  lace  can  be  bought 
for  $1.00?     $.83f?     $1.08J?     $.33J?     $  .66}  ?     $.50? 

61.   Find  gain  and  ratio  of  gain  to  cost. 

a          b            c          d         e         f  g 

Cost,                25?     50^     16}^  33  ^  66|^  75^  91}^ 

Selling  price,  33^   66|^   25^     50^     75^     83^  $1.00 

52.  How  much  would  Mr.  Lee  gain  by  buying  12  yd.  of 
cassimere,  at  $  .66|  a  yard,  and  selling  it  at  $  1.00  a  yard  ? 

53.  Would  he  gain  or  lose,  and  how  much,  by  buying  12  yd. 
of  silk  at  $  .33^  per  yard,  selling  half  of  it  at  $  .50  per  yard, 
and  the  rest  at  $  .25  ? 

54.  -^of  100%=? 

55.  Complete  the  following  table  and  learn  it  : 

=  -         58|%  =  - 


=  —  =  — 

25%   =—         50%   =—         75%   =—        100%   =  — 

56.  Copy  the  six-pointed  star  given  on  page  92  and  divide 
it  into  6  equal  rhombuses. 

57.  If  the  perimeter  of  the  star  were  100  in.,  how  long 
would  the  perimeter  of  one  of  the  rhombuses  be  ? 


204 


ALIQUOT  PARTS 


58.  What  is  the  ratio  of  each  rhombus  to  the  star  ?  Express 
the  ratio  in  per  cent.  What  per  cent  of  the  star  would  remain 
if  one  rhombus  were  erased? 


59.  Draw  a  line  from  each  point  of  the 
star  to  the  center  of  the  figure  as  in  Fig.  1. 
Into  what  kind  of  figures  is  each  rhombus 
divided  ? 

60.  What  is  the  ratio  of  each  isosceles 
triangle  to  the  rhombus  of  which  it  is  a 
part  ?     To  the  star  ? 

61.  If  one  triangle  were  shaded,  what 
per  cent  of  the  star  would  be  unshaded  ? 


FIG.  1. 


62.  Place  letters  at  the  center  and  at  the  vertices  of  the 
angles  of  the  star  and  name  a  figure  that  is  50%  of  the  star. 
75%.  81%.  16|%.  331%.  66|%.  831%. 


63.  Kate  drew  a  square  foot  on  the  board  and  marked  it  off 
into  inches.  She  erased  81%  of  it.  How  many  square  inches 
were  left  ? 

64  o    How  many  cubic  inches  in  8^%  of  a  cubic  foot  ? 

65.    To  gain  81%,  at  what  price  must  goods  be  sold  that  cost 

48^?     60^?     30^?     45^?     72^?     75^?     $1.08? 


66.  Mr.  Barnett  drew  out  of  the  bank  $  56,  which  was  16f  % 
of  the  money  he  had  in  the  bank.  How  much  had  he  in  the 
bank  before  he  drew  any  out  ?  How  much  afterward  ? 


67.    A  line  a  yard  long  was  lengthened 
it  then  ? 


How  long  was 


68.  How  many  times  is  81  contained  in  the  second  multiple 
of!6|?     In  the  5th?    3d?    6th? 

69.  How  many  times  is  3J  contained  in  the  second  multiple 
of!6|?     In  the  3d?     6th?    4th? 


Selling  Price 

Cost 

Selling  Price 

$.66f 

/  ' 

$.831 

$.66f 

.50 

9 

.66| 

•33i 

1.00 

h 

.33J 

.16| 

.66| 

i 

.66| 

1.00 

.50 

j 

.331 

.66| 

h  is  i  of  100  ? 

f?    .*? 

J? 

ALIQUOT  PARTS  205 

70.  Find  ratio  of  gain  or  loss  to  cost  of  goods  at  the  follow- 
ing prices.     Express  ratios  in  per  cent. 

Cost 

a  $.50 
6       .331 
c       .831 
d     1.00 
e       .66| 

71.  How  m 

72.  Beginning  at  121,  name  all  the  multiples  of  121  that  are 
less  than  101.     Write  them  in  a  vertical  column. 

SUGGESTION   TO   TEACHER.      Exercises    should    be    given    upon    12| 
similar  to  those  in  Exs.  39-43  upon  16f . 

73.  Name  all  the  multiples  of  12J,  less  than  101,  that  are 
also  multiples  of  8  ls,  and  tell  how  many  times  they  contain  8^ ; 
also  how  many  times  they  contain  121. 

74.  If  6  yd.  of  calico  are  required  for  Mary's  dress,  how 
much  more  would  it  cost  at  12J^  per  yard  than  at  8J^  ? 

75.  Jennie  has  $  1.25  to  spend  in  ribbon.     How  many  more 
yards  can  she  buy  at  8|^  per  yard  than  at  121^? 

76.  Find  the  amount  of  gain  on  each  article,  and  the  ratio 
of  the  gain  to  the  cost  of  the  goods  at  the  following  prices : 

a          b  c  d          e  f          9 

Cost,  25^     50/     62$  j     75?     12j^     50^     62$  f 

Selling  price,  37$  f   62$  t    87^     S7$t   25^       S7$f    750 

77.  How  much  did  Mr.  Luce  gain  on  14  yd.  of  silk  bought 
at  $  .871  per  yd.  and  sold  for  f  1.25  per  yd.  ? 

78.  Count  from  0  to  100  and  back  from  100  to  0  by  inter- 
vals of  6J. 

SUGGESTION  TO  TEACHER.    Exercises  should  be  given  upon  6£  similar 
to  those  in  Exs.  39-43  upon  1C|. 


206  ALIQUOT  PARTS 

79.  CLASS  EXERCISE.     The  multiples  of  6J  being  written 

upon  the  board, points  to  one  of  them,  and  the  class  give 

quickly  its  ratio  to  6J. 

80.  At  6J  cents  a  yard,  what  is  the  cost  of  8  yd.  of  calico  ? 
12yd.?     3yd.?     5yd.?     7yd.?     9yd.?     llyd.?     6yd.? 

81.  At  6J  cents  a  yard,  how  many  yards  of  calico  can  be 
bought  for  25^?     50^?     $1?     $.12|?     $.37|? 

82.  Find  gain  and  ratio  of  gain  to  cost : 

a  bed  e  f 

Cost,  121^      25^      25^      37^      50^ 

Selling  price,  18|/      3l±f    37^    50^        56^ 

83.  Find  loss  and  ratio  of  loss  to  cost : 

a  be  d         e          f 

Cost,  18|^     25^     37^     50  f     75?     68f^ 

Selling  price,  121^     18|^  31 J^     43f^  68|^  62^     66 J^ 

84.  How  many  times  is  6J  contained  in  the  4th  multiple  of 
12£  ?    In  the  7th  multiple  of  12 J  ?    In  the  3d  multiple  of  12  J  ? 

85.  How  many  more  yards  of  goods  can  be  bought  for  $  1.00 
when  the  price  is  6^  than  when  it  is  121^? 

86.  How  much  did  Mr.   Colton  gain  on  8  yd.  of  denim, 
bought  at  $  .12£  a  yard  and  sold  at  $  .18f  ? 

87.  How  much  did  he  lose  on  20  yd.  of  calico,  bought  at 
$  .06J  a  yard  and  sold  at  $  .05  a  yard  ? 

88.  Find  loss  or  gain  and  its  ratio  to  cost : 

a  b  c  d         e         f          g 

Cost,  66^    37 J^    62^    75  f    25?    43f^ 

Selling  price,   62J^    43|^    68fX    500    12^50^ 

89.  How  much  did  Mr.  Hale  gain  on  8  yd.  of  ribbon,  bought 
at  f  .37J  a  yard  and  sold  at  $  .50  a  yard  ? 


ALIQUOT   PARTS  207 

90.  Write  all  of  the  first  12  multiples  of  8J  that  are  also 
multiples  of  6J,  and  tell  how  many  times  each  of  them  con- 
tains 8^,  and  how  many  times  it  contains  6^. 

91.  How  many  more  yards  of  ribbon  can  be  bought  at  6J^ 
than  at  8J  £  if  the  sum  spent  is  $  1.00  ?  $  .50  ?   $  .75  ?   $  .25  ? 

92    8?x6i  =  ?  93 

' 


25  x  25  16|  x  10  x  25 

2ix3jx6jx8i_?          9g    8txl2jx2t     ? 

'   5xlOx25xl()|  '   33£x6ix7i 

12*x8i-x2^_?  8jx25xl2j-_9 

7£x25x6i  '   25x75x62£~ 

75x25x2j     ?  Q9      871x5x3   _? 

62^x6x5  '    14x12^x2^ 

100.  Draw  the  line  AB  of  any  con- 
venient  length  and  bisect  it. 

To  bisect  the  line  AB.  With  A  as  a 
center  and  any  convenient  radius 
greater  than  one  half  of  AB,  describe 
ah  arc.  With  B  as  a  center  and  the 
same  radius  describe  an  arc  cutting 
the  first  arc  at  C  and  D.  Join  O 
and  D.  The  point  E  where  CD  cuts 
AB  is  the  middle  point  of  the  line 
AB.  CD  is  perpendicular  to  AB. 

101.  Bisect  AE  and  mark  its  middle  point  F.     Bisect  EB 
and  mark  its  middle  point  G. 

102.  Mark  on  each  new  division  what  per  cent  it  equals  of 
the  line  AB. 

103.  What  per  cent  of  the  line  AB  is  AF?    AG?     GB? 

EB?    FG?     FB? 

AxFyEzGmB 

104.    Bisect  each  division  of  the  line 


on  each  division  its  per  cent  of  the  whole  line  AB. 


208  ALIQUOT  PARTS 

105.  What  per  cent  of  the  whole  line  is  Ay?     Am?    Az? 
Fz?     xy?     xm?    Fm?    xB?     yB?    ym?     xz? 

106.  Draw  a   regular  octagon  and  divide  it  into  8  equal 
isosceles   triangles.     Each   triangle  is  what  per  cent   of  the 
octagon  ? 

107.  Divide   each    triangle    into    2    equal    right    triangles. 
Each  triangle  is  what  per  cent  of  the  octagon  ? 

108.  Complete  the  following  table  and  learn  it: 


=—        87i%=  — 
18f%=—        43|%=-          68f%=—        93f%=- 
25%=—          50%=—          75%=—        100%=- 

109.  How  many  square  inches  in  12  1%  of  a  rectangle  1  ft. 
long  and  J  ft.  wide  ? 

110.  How  much  is  37-1%  of  72**  ?     Of  $  1.44  ?"    Of  $  4.80  ? 

111.  To  gain  121%,  how  must  goods  be  sold  that  cost  8^  ? 
40^?     60^?     50^?     $1.20?     $1.60?     $3.20?     $6? 

112.  $5  is  12^-%  of  Charles's  weekly  salary.     How  much 
does  he  receive  each  week? 

113.  Mr.  Owen  sold  15  acres  of  land,  which  was  12-J%  of  his 
farm.     How  large  was  his  farm  before  the  sale  ?    After  the  sale  ? 

114.  In  Mrs.  Abbot's  parlor  there  is  a  rug  6  ft.  long  and  5  ft. 
wide,  which  covers  12^%  of  the  floor.     How  many  square  feet 
are  there  in  the  floor  ? 

115.  What  is  the  ratio  of  the  uncovered  part  of  the  floor  to 
the  whole  floor?     Express  that  ratio  in  per  cent. 

116.  Find  loss  or  gain  and  ratio  to  cost: 

Cost  Selling  Price                            Cost  Selling  Price 

a   $.25  $  .37£  e   $1.00  $  .75^ 

b      .50  .62£  /      1.00  .62£ 

c       .75  .87*  g        .75  .62* 

d       .87i  1.00  h        .37*  .12J 


ALIQUOT  PARTS  209 

By  observing  some  aliquot  parts  of  numbers  we  may  find  short  methods 
of  multiplying. 

117.  Annex  two  ciphers  to  48  and  divide  the  result  by  4. 
Compare  the  result  with  the  product  of  48  x  25. 

Observe  that  by  annexing  two  ciphers  to  any  integer  we  multiply  it  by 
100,  which  gives  a  result  4  times  as  great  as  when  the  integer  is  multiplied 
by  25. 

118.  Take   Ex.    117,    substituting   other   numbers    for    48. 
What  general  truth  connected  with  that  work  can  you  state  ? 

119.  Illustrate   the   following  rules   and   give  reasons   for 
them: 

To  multiply  by  3 J.  —  Annex  a  cipher  to  the  multiplicand  and 
divide  the  result  by  3. 

To  multiply  by  33^.  —  Annex  two  ciphers  to  the  multiplicand 
and  divide  the  result  by  3. 

120.  Tell  how  you  would  multiply  a  number  by  333J  by  the 
method  of  aliquot  parts. 

121.  Multiply  72  by  25.     By  33£.     By  333J. 

122.  Annex  a  cipher  to  36  and  divide  the  result  by  4.  Why 
is  the  quotient  thus  obtained  equal  to  the  product  of  36  and  2|-  ? 

123.  If  you  annex  a  cipher  to  24  and  divide  the  result  by  3, 
the  quotient  thus  obtained  equals  how  many  times  24  ?     Why  ? 

124.  By  2J  multiply  84.     56.     128.     31. 

125.  By  3J  multiply  27.     81.     43.     28.     15. 

126.  Give   a  rule  for  finding  the  product  of  2J  and  any 
integer  without  using  either  2J  or  the  integer  as  a  multiplier. 
Give  reasons  for  the  rule. 

127.  Give  a  similar  rule  for  finding  the  product  of  3J  and 
any  integer.     Give  reasons. 

128.  How  many  ciphers  must  be  annexed  to  an  integer,  and 
by  what  must  the  result  be  divided,  in  order  to  multiply  the 
integer  by  16 J?     By  12£?     Illustrate. 

HORN.  GRAM.   8CH.   AR.  14 


210  ALIQUOT  PARTS 

MISCELLANEOUS    EXERCISES 

1.  What  number  is  6  more  than  1^ 

2.  Square  2.1.     3.2.     1.25.     1.02.     2.003. 

3.  Cube  .02.     .8.     1.2.     1.03.     1.05. 

4.  Change  \  and  f  to  decimals  and  find  their  product. 

5.  Change  to  decimals  and  find  their  continued  product  : 

a  b  c 

*    i    i  if*  iti 

6.  Find  sum  of  all  the  prime  numbers  between  20  and  50. 

7.  How  many  times  4.37  equal  17.48  ? 

8.  What  number  besides  137  will  exactly  divide  11,371  ? 

9.  |  of  8J  x  |  X  ff  X  if  of  if  of  rffr  =  ? 

10.  Reduce  : 

15  10  16|  33|  40  13J 

H  3*  3J  3J  3J  3£ 

11.  A  man  worked  5  da.  at  the  rate  of  $  9f  a  week  (6  da.). 
How  much  did  he  receive  for  the  5  da.  work  ? 


12.  6|xifxAxH=?        5|-TV-ll 

13.  Give  a  fraction  whose  reciprocal  is  less  than  the  fraction 
itself.     Is  the  original  fraction  proper  or  improper?     What 
kind  of  a  fraction  is  the  reciprocal  ? 

14.  Find  33^%   of  the  largest   odd   number  that  can  be 
written  with  two  figures. 

15.  Find  11%  of  the  only  prime  number  between  89  and  101. 

16.  Find  16f  %  of  242. 

17.  Find  25%  of  V64.     Of  Vl44. 

18.  Find  66f  %  of  63.     Of  83. 

19.  Find  50  %  of  the  cube  root  of  the  following  numbers  : 
27        64        343        125        216        729        1728 


MISCELLANEOUS   EXERCISES  211 

20.    Divide  twenty-four  thousandths  by  sixteen  millionths. 


21c    A  lady  buys  a  dollar's  worth  of  soap  at  6J^  a  bar.     If 
she  uses  8  bars  in  1  mo.,  how  long  will  it  last  ? 

22.  How  long  is  the  circumference  of  a  circle  whose  diameter 
is  1^-  cm.  ?     The  arc  that  is  T4T  of  that  circumference  ? 

23.  At  12|^  a  qt.,  how  much  will  2  gal.  of  molasses  cost? 
8  gal.  ?     3  gal.  3  qt.  ?     1  pt.  ?     2  qt.  1  pt.  ?     1  gi.  ?     3  gi.  ? 

24.  It  costs  6  \  ?  to  make  a  gill  of  a  certain  kind  of  medicine. 
If  it  is  sold  for  a  dollar  a  quart,  how  much  is  gained  on  a  quart  ? 

25.  Express  in  per  cent  the  ratio  of  a  peck  to  a  bushel. 

26.  Express  in  per  cent  the  ratio  of  7  m.  to  a  dm.     6  m. 
to  a  Hm.     A  square  meter  to  a  hectare.     9  sq.  dm.  to  a  sq.  m. 
700  cu.  dm.  to  a  stere. 

27.  Reduce  to  a  common  fraction  in  its  lowest  terms  .075. 
.875.     .5625. 

28.  Write  five  fractions  whose  numerators  are  each  6^  and 
whose  denominators  are  multiples  of  6^,  and  reduce  them  to 
lowest  terms. 

29.  Write  six  fractions  whose  numerators  are  each  8J  and 
whose  denominators  are  multiples  of  8J,  and  reduce  them  to 
lowest  terms. 

30.  Divide  50%  of  216.48  by  33. 

177.76-25%  of  177.76  _  9 
1.2 

32.  Change  to  hundredths  expressed  as  a  decimal: 

A      I      «      «      A 

33.  Express  in  per  cent  J.     •}.     f  .     f.     f.     f  .     J.     f  . 

34.  A  regiment  in  marching  takes  128  steps  in  a  minute, 
each  step  2|  ft.  long.     How  many  feet  does  it  advance  in  an 
hour? 


212  ALIQUOT  PARTS 

35.  How  many  rods  of  fencing  will  it  take  to  inclose  a  lot 
45  rd.  3  yd.  long  and  30  rd.  1  yd.  wide  ? 

36.  A  room  is  17f  ft.  long  and  12f  ft.  wide.     What  will  be 
the  cost  of  a  molding  around  it  at  3|^  per  foot  ? 

37.  Divide  the  snni  of  J  and  J  by  -J-. 

38.  A  farmer's  wife  sold  to  a  grocer  30  doz.  eggs  at  18f  ^ 
per  dozen,  receiving  in  payment  a  barrel  of  flour  at  $  5.50  per 
barrel,  and  the  balance  in  cash.     How  much   cash  did  she 
receive  ? 

39.  A  grocer  bought  apples  at  $  1.50  per  bushel,  and  sold 
them  at  50^  a  peck.     How  much  did  he  gain  on  each  bushel  ? 

40.  What  is  the  cost  of  excavating  437.24  cu.  yd.  of  earth 
at  $  1.65  a  cubic  yard  ? 

41.  A  man  sold  J  of  his  farm  of  216  A.  to  one  neighbor, 
and  f  of  it  to  another.     How  many  acres  were  left  ? 

42.  Four  men,  A,  B,  C,  and  D,  together  bought  a  ship  for 
$  16,256.     A  paid  $  4756,  B  paid  $  763  more  than  A,  and  C 
paid  $  256  less  than  B.     How  much  did  D  pay  ? 

43.  How  many  quarts  of  water  can  be  poured  into  a  tin  box 
that  is  11  in.  long,  6  in.  wide,   and   7  in.  deep?      (231  cu. 
in.  =  1  gal.) 

44.  Seven  boys  pick  4  bu.  3  pk.  7  qt.  of  berries  and  share 
them  equally.     What  is  each  boy's  share  ? 

45.  How  many  more  pounds  of  candy  can  be  bought  for  $  1, 
at  6^  per  pound,  than  at  33^  ? 

46.  How  many  axes,  each  weighing  3  Ib.  8  oz.,  can  be  made 
from  a  ton  of  iron  ? 

47.  If  a  man  walks  65  ft.  in  1  min.,  how  many  miles  will 
he  walk  in  10    hr.? 


48.    How  many  steps,  each  2  ft.  6  in.  long,  will  a  boy  take 
in  going  around  a  lot  5  ft.  square  ? 


MISCELLANEOUS  EXERCISES  213 

49.  A  man  has  285  bu.  3  pk.  6  qt.  of  grain,  which  he  wishes 
to  take  to  market  in  15   equal  loads.     How  much  must  he 
put  into  each  load  ? 

50.  A  worker  in  a  cotton  mill  weaves  6  cuts  of  cloth  in  a 
day,  receiving  16f  ^  a  cut.     How  much  does  she  earn  in  a 
week? 

51.  The  circumference  of  a  bicycle  wheel  is  3  ft.     How 
many  times  does  it  turn  in  running  18  ft.?     15  yd.?     4  rd.? 

52.  If  a  horse  eats  2  bales  of  hay,  costing  60^  a  bale,  and 
2  bushels  of  oats,  costing  30^  a  bushel,  in  1  week,  how  much 
does  it  cost  to  feed  him  1  year  of  52|  weeks  ? 

53.  An  automobile  ran  67  f  mi.  in  one  day,  and  1J  times  as 
far  the  next  day.     How  far  did  it  run  in  both  days  ? 

54.  If  Lucy  washes  dishes  3  times  a  day,  how  many  times 
will  she  wash  dishes  in  the  winter  months,  beginning  Dec.  1, 
1903? 

55.  One  of  the  parallel  sides  of  a  trapezoid  is  9T5^  ft.  long. 
The  other  is  3^  ft.  longer.     Each  of  the  non-parallel  sides  is 
5_^.  ft.  long.     How  long  is  the  perimeter  of  the  trapezoid  ? 

56.  Multiply  26  by  11  by  a  short  method. 

This  can  be  done  by  writing  the  sum  of  the  digits  between  them. 
Eight  is  the  sum  of  the  digits  2  and  6.  This  written  between  them  gives 
the  number  286,  which  is  the  product  of  26  and  11. 

57.  By  11  multiply : 

33      72      54      81      45      71      60      22      70 

58.  Think  of  a  number  the  sum  of  whose  digits  is  less  than 
10.     Multiply  it  by  11  by  the  ordinary  written  method,  and 
try  to  discover  why  the  short  method  gives  the  same  result. 

59.  Try  to  discover  how  the  product  should  be  written  when 
the  sum  of  the  digits  is  more  than  9. 

60.  By  11  multiply  : 

36      48      79      85      91      87      58      69      75 


214  ALIQUOT  PARTS 

61.  A  man  had  two  plots  of  land  fronting  a  street.     The  first 
piece  was  600  ft.  wide,  the  second  900  ft.  wide.     He  divided 
them  into  house  lots  of  the  greatest  possible  equal  width. 
How  wide  was  each  lot  ?     How  many  lots  were  there  ? 

62.  A  man  hoed  a  piece  of  land  in  9|  da.,  hoeing  f  of  an 
acre  each  day.     How  many  acres  were  there  in  the  piece  ? 

63.  If  18  suits  of  clothing  can  be  made  from  101  yd.  of 
cloth,  how  many  yards  will  be  needed  for  30  suits  ? 

64.  Mr.  Wade,  dying,  left  $  9600   to   be   divided  equally 
among   his   three   sons,   Arthur,   Henry,   and   Joseph.     How 
much  did  each  receive  ? 

65.  Mr.  Arthur  Wade  bought  a  lot  for  $  1000,  built  a  house 
costing  $  2200,  and  rented  it  for  $  24  a  month.     The  taxes  on 
the  property  for  the  first  year  were  $  75.     The  insurance  was 
$  8.     The  house  was  vacant  two  months.     At  the  end  of  the 
year  he  sold  the  property  for  $  2800.     Did  he  gain  or  lose,  and 
how  much  ? 

66.  Mr.  Henry  Wade  bought  $  3200  worth  of  mining  stock, 
on  which  his  taxes  were  $  13.75.     He  visited  the  mine  at  an 
expense  of  $  22.50.     His  yearly  profits  were  $  173.75.     At  the 
end  of  the  year  he  sold  his  mining  stock  for  $  3000.     Did  he 
gain  or  lose,  and  how  much  ? 

67.  Mr.  Joseph  Wade  bought  a  farm  of  50  A.  at  $40  an 
acre,  with  a  house  worth   $  1200.      The  taxes  on  the  farm 
were  $  41.50.     He  rented  the  farm,  receiving  as  rent  £  of  the 
value  of  the  crops.     The  crops  sold  for  $  935.75.     At  the  end 
of  the  year  the  farm  was  worth  as  much  as  at  the  beginning. 
Did  he  gain  or  lose,  and  how  much  ? 

68.  Scott  &  Co.  of  St.  Louis  bought  a  bill  of  goods  in  New 
York  amounting  to  f  836.75.     They  bought   them  with  the 
agreement  that  they  need  not  pay  for  them  for  three  months, 


MISCELLANEOUS   EXERCISES  215 

but  that  if  they  chose  to  pay  for  them  at  once,  5%  of  the  bill 
would  be  taken  off.  They  chose  to  pay  at  once.  How  much 
did  they  pay  ? 

69.  The  same  firm  bought  another  bill  of  goods  amounting 
to  $  1573.84,  and  were  allowed  a  discount  of  5%   for  cash 
payment.     How  much  did  they  pay  ? 

70.  Kinkle,  Barbour  &  Co.  of  Springfield,  111.,  sold  a  bill  of 
goods  to  Luther  Johnson  of  Jackson,  Tenn.,  as  follows : 

1  doz.  pairs  Men's  Shoes  @  $  1.50  per  pair. 

1  case  (12  pairs)  Children's  Oxfords  @  $  75. 
4  gross  of  Leather  Laces  at  $  .50  per  gross. 

2  cases  Children's  and  Misses'  Oil  Grain  @  $  1.00  per  pair. 
1  doz.  Men's  Kangaroo  Calf  @  $  1.60  per  pair. 

%  doz.  Misses'  Sandals  @  $  .90  per  dozen. 

Make   out   the   bill  and   receipt   it,   allowing    5%    off   for 
cash. 

71.  Mr.  Lang  borrowed  $700  at  7%.     At  the   end  of  3} 
years  how  much  did  he  owe  ? 

72.  Mr.  Lang  borrowed  $800  at  5%  and  at  the  end  of  2 
years  made  a  payment  of  $  200.     How  much  did  he  still  owe  ? 

73.  Mr.  Davis  invested  $625  in  a  mine,  gained  7%  on  his 
investment,  and  paid  $  5.25  in  taxes  on  it.     How  much  of  his 
profit  remained  ? 

74.  Find  loss  and  ratio  of  loss  to  cost  of  goods : 

a        b  c  d        e  f         g         hi 

Bought  at    6^    12?  8^  8^  16^  20^  30^  25^ 

Sold  at        5^      9^  7^  5^  14^  15^  25^  20^ 
Express  in  per  cent. 


CHAPTER   VII 

PERCENTAGE 

1.  The  Latin  phrase  "per  centum,"  meaning  "by  the  hun- 
dred," is  shortened  to  "per  cent,"  and  is  represented  by  the 
sign  "  %." 

Express  15%  as  a  common  fraction.     As  a  decimal. 

2.  Arrange  the  following  in  the  order  of  their  size,  placing 
the  smallest  first  :    T3^.     .33.     34%. 

3.  Express  21%  in  three  different  ways. 

4.  Find  7%  of  $  20  by  the  following  rule  : 
To  find  any  per  cent  of  a  number  — 

Multiply  the  number  by  the  per  cent  expressed  as  a  decimal. 

5.  Give  reasons  for  the  rule. 

6.  By  this  rule  find  11%  of  $48.     Of  96. 

7.  By  the  rule  you  have  been  using  find  50%  of  8.     Show 
an  easier,  way  of  finding  50%  of  8. 

8.  Whenever  a  given  per  cent  can  be  expressed  as  a  common 
fraction,  a  convenient  method  of  finding  that  per  cent  of  a 
given  number  is  to  multiply  the  number  by  that  fraction  ex- 
pressed in  its  lowest  terms. 

In  this  way  find  16f  %  of  1200.     Of  2400.     Of  72.6.     Of  f 

9.  Give  the  value  of  each  of  the  following  fractions  in  per 
cent: 


216 


PERCENTAGE  217 

10.  How  much  is  100%  of  a  dollar?     Of  a  day?     If  you 
had  only  a  dollar,  could  you  spend  150%  of  it  ?     Explain. 

11.  What  is  100%  of  5  books  ?     100%  of  2  watermelons  ? 

12.  The  number  or  quantity  of  which  a  per  cent  is  taken  is 
call  the  Base  of  percentage. 

With  72  as  a  base  find  111%.     444%.     88|%.     16f%. 

13.  Of  a  school  of  40  children  20%   were  absent.     How 
many  pupils  were  absent  ?     How  many  were  present  ?     What 
per  cent  of  them  were  present  ? 

14.  When  you  have  written  40%  of  a  spelling  lesson  of  20 
words,  how  many  words  have  you  written  ?     How  many  are 
yet  to  be  written  ? 

15.  Think  of   $1000;   decrease  it  10%,  then  increase  the 
remainder  33^%,  decrease   that  result  25%,  decrease  $  100, 
increase  50%,  decrease   $200,  increase   50%.     What  is   the 
last  result? 

16.  How  many  different  bases  of  percentage  were  given  in 
the  preceding  problem  ?     Name  each. 

17.  Mr.  Smith  had  $500,  gained  20%,  then  lost  50%  of 
what  he  had,  gained  33£%,  lost  25%,  gained  $300,  lost  16f  %. 
How  much  had  he  then  ? 

18.  Mr.  A.  had  $600,  lost  16|%,  gained  25%,  lost  $25, 
gained  33J%,  gained  121%,  lost  331%,  gained  10%,  lost  $160, 
gained  25%.     How  much  more  had  he  than  at  first^? 

19.  Make  a  problem  similar  to  the  above. 

20.  A  laborer  was  earning  $  1.60  a  day  when  his  wages  were 
cut  25%.     What  were  his  daily  wages  then  ?     After  a  time  his 
wages  were  raised  25%.     How  much  less  wages  did  he  receive 
then  than  he  received  before  the  reduction  ? 

21.  If  the  rate  of  decrease  and  the  rate  of  increase  were 
each  25%,  why  do  they  not  balance  ? 


218  PERCENTAGE 

22.  Tell  which  of  the  following  per  cents  are  most  easily 
found  by  reducing  to  a  common  fraction : 

a  b  c  d  e  f  g 

31  %  16f%  16J%  37i%  11|%  35%  25% 

h  i '•  j  k  I  m  n 

871%  26  %  27  %  55f%  831%  85%  88f% 

23.  Use  144  as  a  base  with  each  of  the  rates  given  in 
Ex.  22. 

24.  Find  101%  of  135.     Of  146.     Of  168.     Of  179. 

25.  We  may  find  10%  of  75  by  placing  a  decimal  point  be- 
tween the  7  and  the  5.     Explain. 

26.  Find  in  this  way  10%  of  25.     Of  88.     Of  $  17.50. 

27.  A  cavalry  regiment  took  960  horses  into  a  battle.     If 
8J%  were  killed,  121%  were  wounded,  and  16f  %  were  caught 
by  the  enemy,  how  many  were  left  for  use  ? 

28.  Mr.  Ellis  willed  25%  of  his  property  to  his  son,  62£% 
to  his  daughter,  and  the  remainder  to  a  library.     After  the 
estate  was  settled  there  remained  $  153,600.     How  much  did 
each  legatee  receive? 

29.  Express  each  of  the  following  ratios,  first  as  a  common 
fraction,  then  as  hundredths  in   decimal  form,  and  then  as 
per  cent: 

3:6     7:10     8:40     1:11     9:72     6:24     24:36     36:48 

30.  What  per  cent  of  35  is  18  ? 

SOLUTION.,    18  equals  $f  of  35.     35  =  100%  of  35.    £f  of  100%  = 

i|xW  =  ^,or51f     Ans.  61»%. 
P&  i 

7 

31.  Illustrate  the  following  rule : 

To  find  what  per  cent  one  number  is  of  another  — 
Express  the  ratio  of  the  one  to  the  other  as  a  common  fraction, 
and  multiply  that  fraction  by  100. 

32.  What  per  cent  of  72  is  12  ?     9  ?     24  ?     36  ?     48  ?     72  ? 


PERCENTAGE  219 

33.  Mary  is  7  yr.  old,  and  her  brother  is  21  yr.  old.     Mary's 
age  equals  what  per  cent  of  her  brother's  age  ? 

34.  Find  what  per  cent  each  number  in  the  first 
column  is  of  each  number  in  the  second  column. 

35.  Find  what  per  cent  each  number  in  the  second      g       25 
column  is  of  each  number  in  the  first  column.  g       4Q 

36.  What  per  cent  of  25  is  each  number  greater    10      18 
than  10  and  less  than  20  ?  4      16 

37.  Find  what  per  cent  9  is  of  each  number  between  20  and 
30.     Between  2  and  12. 

38.  John  bought  and  sold  peanuts,  gaining  4?  on  every  6? 
that  he  invested.     What  per  cent  did  he  gain? 

39.  What  per  cent  is  gained  on  goods  bought  at  12?  and 
sold  ut  14??    Sold  at  15??    16??    18??    20??    21??    22?? 

40.  What  per  cent  is  gained  on  goods : 

abode  f  9 

BoughtatlO^?     11??      8??     15??     5??  16??  16?? 

Sold  at      15??     14??     15??     18??     71??  25??  32?? 

41.  What  per  cent  is  lost  on  goods  : 

a  b  c  d  e  f  g 

Bought  at  20??     18??     25??     15??     7??     10??     10?? 
Sold  at       15??     16??     20??     10??     5??      7??      6?? 

42.  Some  pupils  were  finding  out  facts  about  evaporation. 
They  placed  a  quart  of  water  in  a  shallow  pan  and  left  it  in  the 
sun  until  3  gi.  of  it  had  evaporated.     What  per  cent  was  left  ? 

43.  A  pan  with  straight  sides  was  filled  with  water  to  the 
depth  of  6  in.     The  water  was  left  to  evaporate  until  it  was 
only  5  in.  deep.     What  per  cent  of  the  water  evaporated? 

44.  One  evening  Mrs.  Eaton  prepared  batter  for  buckwheat 
cakes.     She  set  it  to  rise  in  a  jar  10  in.  high,  which  it  half  filled. 
The  next  morning  the  batter  filled  T9^  of  the  space  in  the  jar. 
What  was  the  per  cent  of  increase  ? 


220  PEKCENTAGE 

45.  Russell  caught  12  fish,  and  Walter  caught  8.    What  per 
cent  of  all  the  fish  caught  did  each  boy  catch  ? 

46.  Two  boys  formed  a  partnership  to  mend  bicycle  tires. 
The  material  cost  $5.00,  and  they  received  $7.50.     What  per 
cent  was  gained  ? 

47.  At  8  o'clock  one  morning  Mr.  Field  started  on  a  trip  of 
60  mi.  on  a  train  running  25  mi.  an  hour.     What  per  cent  of 
the  distance  had  he  traveled  by  9  o'clock  ? 

48.  Vincent  and  Fred  bought  a  rabbit  house.     Vincent  paid 
$  1.50,  and  Fred  $  .50.    What  per  cent  of  it  does  each  boy  own  ? 

49.  In  playing  ball,  John  caught  the  ball  23  times  out  of  25. 
What  per  cent  of  the  throws  did  he  miss  ? 

50.  Two  boys,  wishing  to  know  how  much  water  dry  bricks 
would  absorb,  put  30  Ib.  of  brick  into  water.     When  the  bricks 
were  taken  out  they  weighed  32  Ib.      What  per  cent  of  that 
weight  was  water  ? 

51.  Milk  was  poured  into  a  straight  glass  jar  to  the  height 
of  10  in.      The  next  morning  the  cream  in  the  jar  was  ^  in. 
thick.     What  per  cent  of  the  fluid  was  cream  ? 

52.  Mr.  Bates  owed  Mr.  Weber  $  7.50  and  gave  him  a  $  10 
bill  in  payment.     What  per  cent  of  the  ten  dollars  was  returned 
in  change  ? 

53.  What  per  cent  of  a  cubic  foot  are  512  cu.  in.  ?     576  cu. 
in.  ?     824  cu.  in.  ?     1024  cu.  in.  ?     504  cu.  in.  ?     1584  cu.  in.  ? 

54.  What  per  cent  of  $  3.75  is  $  1.50? 

If  the  decimal  point  in  each  term  of  the  fraction  ^—  is  moved  two 

O.75 

places  to  the  right,  the  value  of  the  fraction  is  unchanged.     |f §  can  then 
be  reduced  to  its  lowest  terms,  \. 

20 

f  x  W  =  40.  Ans.  40%. 

P 
It  is  often  more  convenient  not  to  reduce  the  fraction  to  its  lowest 

termS'aS  375)15000(40 

1600  Ans.  40%. 


PERCENTAGE  221 

55.  What  per  cent  of  $62.50  is  $12.50?     $37.50?    $18.75? 
$31.25? 

56.  Mr.  Hull  bought  some  goods  for  $  87.50.      How  much 
would   he    gain   by    selling    them    for    $112.50?       $93.75? 
118.75  ?     $131.25  ?     $  121.875  ? 

57.  Mr.   Gordon  bought  groceries  costing  $  218.75.     What 
per  cent  would  he  gain  or  lose  by  selling  them  for  $  243.75  ? 
$193.75?     $206.25?     $212.50?     $256.25?     $225? 

58.  What  per  cent  of  133£  is  16|  ? 

59.  Write  5  numbers  that  are  multiples  of  8£,  and  find  what 
per  cent  each  one  is  of  116J . 

60.  Write  5  numbers  that  are  multiples  of  3£,  and  find  what 
per  cent  each  one  is  of  150. 

61.  What  per  cent  is  gained  by  buying  goods  at  $  .16  J  a 
yard,  and  selling  them  at  $  .25  per  yard  ?     $  .30  ?     $  .331  ? 

62.  What  per  cent  is  lost  on  goods  by  buying  them  at  $  .75 
and  selling  them  at  $.70?    $.62£?    $.66f?    $.50?    $.56J? 
$.45? 

63.  After  a  line  5  in.  long  was  increased  20%,  what  per 
cent  of  1  ft.  was  its  length  ? 

64.  Edwin  lost  7  ^,  which  was  33J  %  of  his  money.     How 
much  had  he  at  first? 

Consider  33|%  as  \. 

65.  Five  cents  is  25%  of  what  sum  of  money  ?    12|%  ? 
16f%? 

66.  Find  the  number  of  which  15  is  66|%. 
Consider  66f%  as  f. 

67.  66  J%  of  a  railroad  is  in  Missouri.     If  that  part  is  252 
miles  long,  what  is  the  length  of  the  railroad  ? 

68.  35  is  87^%  of  what  number  ?     83^%  of  what  number  ? 


222  PERCENTAGE 

69.  6  is  3%  of  what  number? 

SOLUTION 
MlfeM 

then  ^  =  2 
and  fl$  =  200  Ans. 

70.  By  the  same  reasoning  find  the  number  of  which  8  is 
4%.     The  number  of  which  21  is  3%. 

71.  Tell  of  what  number  I  am  thinking  if  35  is  7%  of  it. 
If  4  is  3%  of  it. 

72.  CLASS  EXERCISE.     may  think  of  a  number  and  tell 

the  class  how  much  a  certain  per  cent  of  it  is.     The  class  may 
find  the  number. 

73.  Can  you  see  the  reason  for  the  following  rule  ? 

To  find  the  number  of  which  a  given  number  is  a  certain  per 
cent  — 

Multiply  the  given  number  by  100  and  divide  the  result  by  the 
number  of  per  cent. 

74.  Sixteen  equals  8%   of  what  number?     40%   of  what 
number  ? 

75.  Find  the  number  of  which  $  24  is  12%.     50%.     15%. 

76.  Find  the  number  of  which  $45.75  is  5%.     15%.     25%. 

77.  How  much   money  has  a  man  if  7%  of  it  is  $3500? 
$  6216  ?     $  3110.80  ?     $  4498.20  ? 

78.  In  a  certain  city,  75%  of  the  telegraph  wires  are  under 
ground.     If  300  mi.  are  under  ground,  how  many  miles  are 
above  ground? 

79.  Mr.  Allen  sold  Mr.  Cummings  1240  bu.  of  corn,  which 
was  62%  of  his  entire  crop.     How  much  was  his  entire  crop? 

80.  During  a  storm  at  sea  600  bu.  of  grain  were  thrown  over- 
board.    How  many  bushels  were  there  on  board  before  the 
storm,  if  the   number  of  bushels  thrown  overboard  equaled 
24%  of  the  whole  number  ? 


PERCENTAGE  223 

81.  Nine  hundred  acres  of  a  Florida  plantation  are  marsh. 
How  many  acres  are  there  in  the  whole  plantation  if  the  marsh 
is  75%  of  it? 

82.  Mr.  Leeds  lost  in  a  business  transaction  $  1850.50,  which 
was  15%  of  his  entire  property.     What  was  the  value  of  his 
property  before  his  loss  ?     After  his  loss  ? 

83.  Sixty  years  is  150  %  of  Mr.  Harvey's  age.    How  old  is  he? 

84.  Eighteen  months  is  75%  of  Edgar's  age.     How  old  is  he? 

85.  Forty  per  cent  of  a  load  of  hay  was  clover.     If  1  T. 
was  clover,  what  was  the  weight  of  the  load  ? 

86.  A  rectangle  9  in.  long  and  8  in.  wide   equals  3%  of 
another  rectangle.     What  is  the  area  of  the  larger  rectangle  ? 

87.  A  rectangle  8  in.  by  6  in.  equals  4%  of  a  rectangle 
which  is  6  ft.  long.     How  wide  is  it  ? 

88.  Selling  a  bicycle  for  $30  more  than  I  gave  for  it,  gives 
me  a  gain  of  25%.     How  much  did  I  give  for  it  ? 

89.  I  gained  20%  by  selling  a  watch  for  $5.75  more  than  I 
paid  for  it.     How  much  did  I  pay  for  it  ? 


90.  I  sold  some  goods  at  a  profit  of  $  418,  which  was 

of  their  cost.     How  much  did  they  cost  me  ?     For  how  much 
did  I  sell  them  ? 

91.  By  selling  a  house  for  $  100  less  than  it  cost  I  lost  5%. 
How  much  did  the  house  cost  me  ? 

92.  By  selling  a  stove  for  $  5.80  less  than  cost,  I  lost  29%. 
What  was  the  cost  ? 

93.  I  gained  $340  by  selling  wheat  at  an  advance  of  20%. 
How  much  did  I  pay  for  the  wheat  ? 

94.  A  merchant  sold  silk  at  $  2.20  per  yard,  which  was  110% 
of  what  it  cost  him.     How  much  did  it  cost  him  ?     What  per 
cent  did  he  gain  on  it  ? 


224  PERCENTAGE 

95.  What  is  the  cost  of  goods  sold  at  18^  a  yard,  which 
is  20%  more  than  the  cost  ? 

SOLUTION.  As  the  selling  price  is  20%  more  than  the  cost,  it  is 
120%  of  the  cost.  If  18  jZ  is  |£$  of  the  cost,  the  cost  is  jf$  or  f  of  18ft 
which  is  15  p. 

96.  Find  the  cost  of  tea  sold  at  85^  per  pound,  which 
was  25%  above  cost. 

97.  Find  the  cost  of  goods  sold  at  the  following  prices, 
which  are  25%  above  cost: 

a  Calico,  @  $  .05  per  yard.  d  Velvet,  @  $  1.85  per  yard. 
b  Ribbon,  @  .65  per  yard.  e  Lace,  @  2.75  per  yard, 
c  Flannel,  @  1.35  per  yard.  /  Silk,  @  4.75  per  yard. 

98.  Thirty  per  cent  less  than  the  cost  of  anything  equals 
what  per  cent  of  its  cost  ?     Illustrate. 

99.  A  wheel  was  sold  for  $  28,  which  was  30%  less  than 
cost.     Find  the  cost. 

100.  A  stove  sold  for  $  6  brought  40%  less  than  cost.     Find 
the  cost. 

101.  Arthur   had   $4.20,  which  was   40%  more  than  his 
brother  had.     How  much  had  his  brother  ? 

102.  Mr.  X  raised  350  bushels  of  wheat  this  year,  which  is 
16f  %  more  than  the  number  of  bushels  that  he  raised  last 
year.     How  much  wheat  did  he  raise  last  year  ? 

103.  A  grocer  has  180  Ib.  of  dry  sugar.     In  drying,  it  lost 
10%  of  its  weight.     What  was  its  weight  before  drying? 

104.  There  are  240  A.  in  Mr.  King's  farm,  which  is  20% 
less  than  the  number  of  acres  in  his  uncle's  farm.     How  many 
acres  has  the  latter  ? 

105.  A  merchant  had  380  yd.  of  cloth  after  it  had  shrunk. 
What  was  its  length  before  shrinking,  if  the  shrinkage  was  5%  ? 


PERCENTAGE  225 

106.  Twenty  per  cent  of  a  pole  was  broken  off.     The  part 
remaining  was  16  ft.  long.     What  was  the  length  of  the  pole 
before  it  was  broken  ? 

107.  After  using  10%  of  a  load  of  coal  I  find  there  are 
900  Ib.  left.     How  many  pounds  were  in  the  load  ? 

108.  In   Mr.  Crosby's   orchard   there  are  336  fruit  trees; 
16f  %  of  them  are  plums,  12-J-%  of  them  are  pear  trees,  and  the 
remainder  are  peach  trees.    How  many  are  there  of  each  kind  ? 

109.  A  man  whose  wages  are  $14  per  week,  spent  8%  of  his 
earnings  last  year  for  music  lessons.     How  much  did  he  spend 
for  the  lessons  if  he  worked  50  weeks  ? 

110.  A  man  buys  a  horse  for  $  200  and  sells  it  for  $  46  more 
than  it  cost  him.     What  per  cent  does  he  gain  ? 

111.  A  bicycle  is  bought  for  $75  and  sold  at  a  gain  of 
$  13.50.     What  per  cent  is  gained  ? 

112.  A  watch  costing  $25  is  sold  at  an  advance  of  $4.75. 
What  per  cent  is  gained  ? 

113.  A  pile  of  wood  50  ft.  long,  6  ft.  wide,  and  5  ft.  high  was 
25%  oak,  33^%  maple,  21%  beech,  and  the  rest  walnut.     How 
many  cords  were  there  of  each  ? 

114.  The  average  lung  capacity  of  men  is  about  320  cu.  in. 
If  your  lungs  hold  75%  as  much,  how  much  will  they  hold? 
If  your  clothing  were  so  tight  as  to  reduce  your  lung  capacity 
40  cu.  in.,  what  per  cent  would  the  reduction  be  ? 

115.  The  amount  of  food  required  by  a  man  is  about  51  Ib.  a 
day.     If  you  ate  80%  as  much,  in  how  many  days  would  you 
eat  one  ton  and  two  pounds  of  food  ? 

116.  A  man's  heart  weighs   about   11   oz.     If  your  heart 
weighs  70%  as  much  as  that,  how  much  does  it  weigh? 

117.  Of  a  certain  fence,  200  rd.  were  wire  fencing,  and  the 
rest  rail  fencing.     If  60  %  was  rail,  what  was  the  length  of  the 
fence  ? 

HORN.    GRAM.    SCH.    AR.  15 


226  PERCENTAGE 

118.  Forty-five  miles  of  country  road  is  macadamized.     If 
this  is   20%   of  the  whole   road,  how  much   remains   to   be 
macadamized  ? 

119.  How  much  will  it  cost  to  plaster  a  room  20  ft.  4  in. 
long,  18  ft.  wide,  8  ft.  high,  at  $  .20  a  square  yard,  if  12£%  of 
the  surface  is  covered  with  wainscoting  ? 

120.  A  certain  school  enrolls  23  boys  and  22  girls.     The 
girls  equal  what  per  cent  of  the  school  ? 

121.  The  same  school   has   an   average   attendance  of  42. 
What  per  cent  of  the  enrollment  is  the  attendance  ? 

122.  What  per  cent  of  the  pupils  in  your  school  are  boys  ? 

123.  If  3  pupils  belonging  to  your  school  were  absent  every 
day,  the  attendance  would  be  what  per  cent  of  the  enrollment  ? 

124.  A  boy  having  broken  his  bicycle  sold  it  for  $  20,  which 
was  33 1  %  of  the  cost.     What  was  the  cost  ? 

125.  I  buy  goods  at  $  1.20  and  sell  them  at  a  gain  of  10%. 
What  is  the  selling  price  ? 

126.  I  sell  goods  at  $1.32,  gaining  10%.     Find  the  cost. 
The  gain. 

127.  I  sell  goods  at  a  gain  of  $  12,  which  is  10%  of  the  cost. 
Find  the  cost. 

128.  I  buy  goods  for  $  120  and  sell  them  for  $  12  more  than 
they  cost  me.     What  per  cent  do  I  gain  ? 

129.  I  buy  goods  for  1 120  and  sell  them  for  $  132.     What 
per  cent  do  I  gain  ? 

130.  Your  age  in  years  is  what  per  cent  of  the  age  of  a  per- 
son who  is  4  yr.  older  than  you  ?     6  yr.  older  ?     10  yr.  older  ? 

131.  A  man  sold  two  horses  for  f  150  each,  gaining  20%  on 
one  and  losing  20%  on  the  other.     Did  he  gain  or  lose  by  the 
two  transactions,  and  how  much  ? 

132.  An  inkstand  that  cost  50  fi  was  sold  at  a  gain  of  100%. 
What  was  the  selling  price  ? 


PERCENTAGE  227 

133.  If  berries  that  cost  5^  a  quart  are  sold  for  10^  a  quart, 
what  is  the  per  cent  of  gain  ? 

134.  How  long  will  it  take  a  dollar  to  gain  100%  if  it  is  at 
interest  at  5%  ?     At  6%  ?     8%  ?     9%  ?     12%  ? 

135.  Walter  and  Thomas  sold  lemonade  at  a  fair.     Walter 
furnished  $  .25  worth  of  lemons,  and  Thomas  $  .50  worth  of 
sugar.     To  what  per  cent  of  the  profits  was  each  boy  entitled  ? 
If  they  took  in  $  2.64,  how  much  was  each  boy's  share  of  the 
gain? 

136.  Three  men  raised  a  fund  for  charity.     One  man  gave 
$  15,  another  $>  20,  another  $  25.     What  per  cent  of  the  whole 
did  each  man  give  ? 

137.  Mr.  Low  had  his  money  invested  in  three  houses  as 
follows  :    in  the  first  $  1620,  in  the  second  $  8100,  in  the  third 
$3240.     What  per  cent  of  his  money  was  invested  in  each 
house  ? 

138.  Mr.  Eves  insured  Mr.  Croft's  building  for  $  1600,  which 
is  75%  of  its  value.     What  is  its  full  value  ?     If  Mr.  Croft 
paid  1%  upon  the  amount  insured,  how  much  did  he  pay  ? 

139.  Mr.  Jones's  yearly  income  from  a  mine  is  $  4000,  which 
is  15%   of  the  sum  he  invested  in  it.     How  much  did  he 
invest  ? 

140.  A  store  is  rented  for  $  60  a  month.     The  yearly  rent  is 
8-J%  of  the  value  of  the  property.     What  is  its  value? 

141.  A  lawyer  collected  some  money  for  his  client,  receiving 
for  his  services  $  80,  which  was  5%  of  the  sum  collected.     How 
much  did  he  collect  and  how  much  did  he  pay  over  to  his 
client  ? 

142.  Mr.  Eoy  gained  8%  by  selling  his  cow  for  $20  more 
than  it  cost  him.     For  how  much  did  he  sell  it  ? 

143.  Mr.  Litch  received  $  500  a  year  rent  for  one  of  his 
houses,  which  was  7  %  of  its  value.     What  was  its  value  ? 


228  PERCENTAGE 

144.  After  Mr.  Lane  had  paid  37£%  of  his  debts,  he  found 
that  $3568  would  pay  the  remainder.      What  was  his  total 
indebtedness  ? 

145.  Mr.  Bingham  sold  two  houses  for  $  7000  each;  for  the 
first  he  received  12|%  less  than  its  value  and  for  the  second 
16f  %  more  than  its  value.     What  was  the  value  of  each  ? 

146.  Mr.  Allen  bought  for  $  3200  a  store  which  had  depre- 
ciated 40%   in  value.     What  was  the  original  value  of  the 
store  ? 

147.  When  the  Twentieth  Century  Novelty  Company  failed 
in  business  they  had  $  3350,  which  paid  67  %  of  their  debts. 
What  was  the  amount  of  their  debts  ? 

148.  A  grocer  sold  his  stock  for  $  2000,  which  was  at  a  loss 
of  10%.     How  much  did  the  stock  cost  ? 

149.  Mr.  Drew  bought  5  doz.  oranges.     Of  these  30%  were 
not  good.     How  many  oranges  were  good  ? 

150.  A  florist   used   in   January  4000  bu.   of   coal.      This 
quantity  was  36%  of  the  number  of  bushels  used  during  the 
winter.     How  many  bushels  were  used  in  all  ? 

151.  Mr.  M.  receives  $150  each  ye.ar  as  the  interest  on  a 
sum  of  money  which  pays  6%.     What  is  the  sum  of  money  ? 

152.  How  much  money  must  one  have  at  interest  that  he 
may  receive  from  it  $  750  a  year  when  the  rate  of  interest 
is  5%  ?     6%  ?     8%  ?     4%  ?     3%  ? 

MERCHANDISING 

153.  Buying  and  selling  goods  for  profit  is  called  Merchan- 
dising.    Those  who  carry  on  merchandising  are  called  mer- 
chants.    Mention  several  lines  of  merchandising. 

154.  A  merchant  bought  $  6000  worth  of  dry  goods,  and  in 
the  first  year  gained  20%  of  his  capital.     How  much  did  he 
gain  that  year  ? 


MERCHANDISING  229 

155.  He  used  $  900  that  year  for  living  expenses,  and  put 
the  rest  into  his  business.     What  was  his  capital  at  the  begin- 
ning of  the  second  year  ? 

156.  In  the  second  year  he  cleared  33-^%  on  his  capital. 
What  was  his  gain  ? 

157.  The  second  year  he  used  $  1200  for  his  living  expenses, 
and  put  the  balance  into  his  business.     What  was  his  capital 
at  the  beginning  of  the  third  year  ? 

158.  In  the  third  year  he  gained  37%%  on  his  capital.     His 
expenses  outside  of  his  business  that  year  were  $  1800.     The 
rest  of  his  gain  went  into  the  business.     What  was  his  capital 
at  the  end  of  the  third  year  ? 

159.  In  the  fourth  year  he  gained  44|-%  on  his  capital,  and 
used  $  1700  for  outside  expenses.     What  was  his  capital  at  the 
end  of  the  fourth  year  ? 

160.  January  1, 1899,  Mr.  A.  went  into  business  with  a  capi- 
tal of  $10,000.     At  the  end  of  the  year  he  found  that  the 
amount  of  his  sales  for  the  year  had  been  $  18,448 ;  that  his 
business  expenses,  such  as  rent,  clerk  hire,  advertising,  etc., 
had  been  $  6000 ;  that  he  had  spent  $  10,625  in  replenishing 
his  stock  of  goods.    How  much  had  he  gained  that  year  ?    His 
living  expenses  were  $1000.     What  was  his  capital  Jan.  1, 
1900,  including  the  amount  invested  in  goods  ? 

161.  In  1900  he  netted  10%  on  his  capital,  and  his  living 
expenses  were  $  1500.     What  was  his  capital  Jan.  1,  1901  ? 

162.  In  1901  his  profits  were  7%  on  his  capital,  and  his 
living  expenses  were  $  1300.     What  was  his  capital  Jan.  1, 
1902? 

SUGGESTION  TO  TEACHER.  Lead  pupils  to  realize  some  of  the  condi- 
tions of  merchandising,  and  let  them  make  similar  problems,  tracing  the 
course  of  simple  business  ventures. 

163.  A  merchant  buys  broadcloth  at  $  2.40  per  yard.     How 
shall  he  mark  it  that  he  may  sell  it  at  a  gain  of  40%  ?    33-|%  ? 
35%?     50%? 


230  PERCENTAGE 

164.  How  much  would  he  gain  by  selling  20  yd.  at  each  of 
those  rates  of  advance  ? 

165.  Mr.  B.  bought  40  yd.  of  novelty  goods  at  $  1.60  per  yard. 
He  sold  ^  of  it  at  an  advance  of  75%,  ^  of  it  at  an  advance  of 
50  °/o  i  and  the  rest  at  25  %  advance.    How  much  did  he  gain  on 
the  whole  piece  ? 

166.  Mr.  C.  bought  60  yd.  of  goods  at  $  1.20  per  yard.     He 
sold  i  of  the  piece  at  a  gain  of  60%,  -i-  of  the  rest  at  a  gain  of 
50%,  14  yd.  at  $  1.30  per  yard,  and  the  rest  at  $.75  per  yard. 
How  much  did  he  gain  on  that  piece  ? 

167.  Mr.  Evans  bought  goods  at  $.75  a  yard  and  marked 
them  to  sell  at  an  advance  of  33J%.     Later  in  the  season  he 
sold  them  at  a  reduction  of  10%  from  the  marked  price.    What 
was  the  actual  selling  price  ? 

Find  actual  selling  price,  and  amount  of  gain  or  loss  on  goods : 

Bought  Marked  Sold 

168.  $.60  50%    above  cost  10%    below  marking 

169.  .80  25%    above  cost          30%    below  marking 

170.  .90  33^%  above  cost          16f  %  below  marking 

171.  1.75  $  2.00  above  cost  12J%  below  marking 

172.  1.60  25%    above  cost  12^%  below  marking 

173.  CLASS  EXERCISE.     may  mention  prices  at  which 

goods  might  be  bought,  marked,  and  sold,  and  the  class  may 
find  the  amount  of  gain  or  loss. 

COMMISSION 

174.  Instead  of  buying  and  selling  for  themselves,  some 
merchants  buy  or  sell  goods  for  others  at  a  given  per  cent  of 
their  value.     The  money  which  they  receive  for  buying  or  sell- 
ing is  called  their  Commission,  and  they  are  called  commission 
merchants,  agents,  or  brokers. 

What  is  the  commission  on  $  900  at  10%  ? 


COMMISSION  231 

175.  Mr.  Ward  sold  a  piano  for  $  575,  receiving  a  commis- 
sion of  10%  on  the  selling  price.     How  much  was  his  com- 
mission, and  how  much  money  should  he  send  to  the  owner  of 
the  piano  ? 

176.  Mr.  Clark  is  an  agent  for  the  Elliott  Piano  Co.  which 
pays  him  a  commission  of  25%  on  all  his  sales.     During  the 
month  of  January,  1899,  he  sold  one  piano  at  $325,  one  at 
$  400,  two  at  $  250,  and  three  at  $  200.     How  much  were  his 
commissions,  and  how  much  should  he  return  to  the  Elliott 
Piano  Co.  ? 

177.  The  expenses  of  his  store  for  rent,  heating,  lighting 
advertising,  clerk  hire,  etc.,  for  January,  were  $  178.50.     How 
much  did  Mr.  Clark  make  that  month  above  expenses  ? 

178.  His   living   expenses   for  that   month  were  $  109.75. 
How  much   did  he  save  ?     Would  it  be  safe  to  reckon  his 
yearly  income  on  this  basis  ?     Why  ? 

179.  A  cotton  broker  receives  a  shipment  of  cotton  from 
Alabama  consisting  of  400  bales.     He  sells  it  at  $  20  a  bale, 
receiving  11%  commission.     How  much  is  his  commission  on 
that  sale,  and  how  much  should  he  return  to  the  owners  of  the 
cotton  ? 

180.  John  sold  his  sister's  sled  for  her,  receiving  $  1  for  it. 
When  he  gave  his  sister  the  dollar,  she  handed  him  a  dime  for 
selling  the  sled.     What  per  cent  was  his  commission  ? 

181.  Mr.  Adams  sold  a  wagon  for  the  Melvin  Wagon  Co.  at 
$  200,  and  after  taking  out  his  commission,  sent  the  Wagon 
Co. -$180.     How  much  was  his  commission?     What  per  cent 
of  -the  value  of  the  wagon  ? 

182.  Mr.  Gordon  buys  chickens  for  a  Baltimore  firm,  re- 
ceiving a  commission  of  15%  on  all  the  money  spent  in  buying 
them.     If  he  buys  $  12,000  worth  of  chickens  in  a  year,  what 
is  his  income  for  that  year  ? 


232  PERCENTAGE 

183.  Mr.  Wilson  travels  in  the  South,  selling  stoves.     He 
receives  a  commission  of  10%  on  sales.     When  he  has  sold 
$  18,000  worth  of  stoves,  how  much  has  he  earned  ? 

184.  Mr.  Wood  buys  hogs  for  a  pork-packing  establishment, 
receiving  a  commission  of  3%  on  the  amount  spent  for  the 
hogs.     If  he  buys  $  7500  worth  in  a  month,  what  are  his  earn- 
ings for  that  month  ? 

185.  A  commission  merchant  bought  $  5000  worth  of  pea- 
nuts in  Tennessee,  and  shipped  them  to  a  Chicago  firm.     The 
cost  of  hauling  them  to  the  depot  in  drays  was  $  2.75,  the 
freight  charges  were  $  47.50.     The  buyer's  commission  was 
4%   on  the  amount  paid  for  the  peanuts.      What  was   the 
amount  of  his  commission  ?     How  much  did  the  peanuts  cost 
the  Chicago  firm  ? 

186.  A  commission   merchant   in  Kentucky  bought  for  a 
New  York  firm  3000  Ib.  of  pecans  at  $  .06^  per  pound.     How 
much  was  his  commission  at  2%  ? 

187.  CLASS  EXERCISE.     may  fill  out  the  following,  and 

the  class  may  find  the  amount  received  as  commission: 

!1   worth  of  r 
sold                  .   .               ^ 
,        ,  ,  I  receiving %  com' 
g      J   mission. 

188.  People  are  sometimes  hired  to  collect  money  due  to 
other  people.      These  collectors  receive  a  percentage  on  the 
amount  collected. 

How  much  does  Mr.  Cox  earn  by  collecting  a  bill  of  $  600  at  2  %  ? 

189.  Mr.  Ho  well  gave  Mr.  Scott  a  list  of  bills  to  be  col- 
lected, 5%  of  the  amount  of  which  Mr.  Scott  was  to  receive  as 
commission.     How  much  did  he  earn  by  collecting  the  follow- 
ing bills,  and  how  much  did  he  turn  over  to  Mr.  Howell  ? 

John  Andrews,  Groceries,  $  28.75. 
Charles  Stockton,  21  Ib.  Sugar,  @  $  .05. 
George  Baldwin,  12£  Ib.  Turkey,  @  $  .15. 
Peter  Garrison,  10  gal.  Molasses,  @  $  .45. 


TRADE   DISCOUNT  233 

190.  CLASS    EXERCISE.  -  may   fill  out  the   following, 
and  the  class  may  find  the  collector's  receipts : 

A  collector  collects ,  receiving  for  his  work %. 

191.  A  collector  undertook  to  collect  $2125  worth  of  bills 
due  a  physician.     He  collected  67%  of  them,  receiving  10% 
as  his  fee.     How  much  did  he  collect  ?     How  much  did  he 
keep  ?     How  much  did  he  send  to  the  physician  ? 

192.  Mr.  Blair  sold  a  carriage  for  the  Hale  Carriage  Co.  at 
$450.     His  commission  was  15%.     The  purchaser  paid  $300 
cash,  and  Mr.  Blair  was  obliged  to  pay  a  collector  10%  on  the 
balance  for  collecting  it.     How  much  did  Mr.  Blair  gain  on 
the  sale  ? 

TKADE  DISCOUNT 

•    193.    A  reduction  in  the  selling  price  of  goods  is  called  a 
Discount. 

Mr.  Reed  found  that  he  could  buy  the  bicycle  that  he  wanted 
for  $  70,  on  3  months'  time,  or  for  $  66.50  cash.  How  much 
was  the  discount  ? 

194.  How  much  is   paid  for  a  bill   of  goods  invoiced  at 
$  18.75,  with  a  discount  of  30%? 

195.  The  employees  of  a  certain  large  dry  goods  store  are 
allowed  to  buy  goods  from  it  at  a  discount  of  10%.     What 
amount  was  paid  for  a  bill  of  $  17.25  bought  by  an  employee  ? 

196.  A  man  received  a  bill  for  $  75.     Printed  on  the  bill 
head  were  the  words,  "5%  discount  if  paid  within  30  days." 
The  bill  was  paid  at  the  end  of  4  weeks.     What  amount  was 
paid? 

197.  Hale  &  Co.,  Springfield,  111.,  bought  of  D.  W.  Lamont 
&  Co.,  St.  Louis,  Mo.,  3  doz.  plain  gold  rings,  @  $  20  per  doz. ; 
4  gold  rings,  diamond  settings,  @  $  50 ;   6  gold   watches,  @ 
$  15 ;  4  sets  teaspoons,  @  $  6.     Make  out  the  bill,  allowing  a 
discount  of  25%. 


234  PERCENTAGE 

198.  Mr.  K.   sold  Mr.  D.  furniture  for  his   home  to  the 
amount  of   $  75,  on  60  days7  time ;  but  Mr.  D.  accepted  the 
offer  of  3J%  off  for  cash.     How  much  did  Mr.  K.  receive  for 
the  furniture  ? 

Many  manufacturers  and  wholesale  dealers  have  a  fixed  price  for  their 
goods,  called  the  list  price.  They  sell  to  the  retail  dealers  at  a  discount 
from  the  list  price.  For  instance,  a  manufacturer  of  carriages  sends  out 
a  catalogue  containing  a  list  of  the  different  kinds  of  carriages  he  makes, 
and  their  prices,  with  the  discounts  he  gives  to  retail  dealers. 

199.  Howard,   Cowperthwait,   &    Co.,   who   sell    carriages, 
select  from  a  manufacturer's  list  a  carriage  whose  list  price 
is  $  750.     The  discount  on  this  carriage  is  55%.     How  much 
do  Howard,  Cowperthwait,  &  Co.  pay  for  the  carriage?     If 
Mrs.  Douglas  buys  the  carriage  for  $  700,  how  much  do  they 
make  on  it  ? 

200.  At  the  same  rate  of  discount,  how  much  will  be  gained* 
by  Howard,  Cowperthwait,  &  Co.  if  they  buy  four  carriages 
whose  list  price  is  $  600,  and  sell  them  at  the  list  price  ? 

201.  Frequently  more  than  one  discount  is  given  upon  the 
same  purchase.     If  Howard,  Cowperthwait,  &  Co.  buy  a  $  1000 
carriage  at  55%   off,  its  cost  is  $  450.     If  they  get  an  ad- 
ditional discount  of  10%,  it  is  reckoned  on  the  $  450  and 
deducted  from  it.     In  that  case,  how  much  will  the  carriage 
cost  them  ? 

Observe  that  when  there  is  more  than  one  discount,  each  successive 
discount  is  made  upon  a  smaller  sum  of  money  than  the  preceding. 

202.  How  much  is  left  of  $800  when  it  is  discounted  25%, 
then  33|%,  then  25%,  then  33£%  ? 

Find  cost  of  the  following : 

List  price  Discount  List  price  Discount 

203.  $  175  45  and  10  off    206.  $  500  20,  10,  and  5  off 

204.  $  370  35  and  5  off    207.  $  600  40,  30,  and  2  off 

205.  $  350  15  and  10  off   208.  $  800  50,  10,  and  5  off 


TRADE   DISCOUNT  235 

209.  If  you  were  buying  goods  should  you  prefer  discounts 
of  15%  and  10%,  or  a  straight  discount  of  25%  ?     Find  the 
discount  on  different  sums  at  these  rates,  and  state  which  terms 
are  more  advantageous  to  the  buyer,  and  why. 

210.  A  dealer  buys  wagons  whose  list  price  is  $  60,  at  a 
discount  of  30%,  20%,  and  5%  off  for  cash.     How  much  do 
these  wagons  cost  him  ?     He  sells  them  at  the  list  price  with 
5%  off  for  cash.     How  much  does  he  make  on  a  cash  sale  ? 
On  a  time  sale  ? 

If  a  time  sale  is  made  in  which  the  time  exceeds  a  certain  limit,  the 
buyer  gives  his  note  for  the  amount  and  pays  the  interest  upon  it.  This 
time  limit  varies  with  different  firms,  being  usually  not  less  than  60  da. , 
nor  more  than  6  mo. 

211.  How  much  is   paid   for  a  bill  of   goods   invoiced   at 
$37.25,  discounts  10,  15,  and  5  off.     How  much  is  gained  if 
the  goods  are  sold  10  %  above  the  list  price  ? 

212.  Ball  &  Co.,  piano  dealers,  select  from   the  manufac- 
turer's catalogue,  3  pianos  listed  at  $400,  $450,  and  $700, 
and  order  6  pianos  of  each  kind.     The  trade  discount  is  60,  10, 
and  5.     What  is  the  cost  of  that  shipment  of  pianos  ? 

213.  Mrs.  Fox  pays  cash  for  one  of  the  $  400  pianos  sold  at 
list  price,  and  gets  a  discount  of  5%.     How  much  do  Ball  & 
Co.  make  on  that  sale  ? 

214.  Mr.  Shaw  buys  a  $700  piano  at  list  price,  pays  $75 
cash,  and  gives  his  note  for  the  balance,  payable  in  6  mo.     At 
the  end  of  that  time,  finding  himself  unable  to  pay  for  it,  he 
returns  the  piano  to  Ball  &  Co.,  who  spend  $  2  in  polishing  and 
tuning  it,  and  then  sell  it  for  $  650.     How  much  do  Ball  &  Co. 
make  on  that  piano  ? 

215.  Mr.  King  buys  one  of  the  pianos  whose  list  price  is 
$  450,  on  the  installment  plan,  paying  $  10  a  month.     Ball  & 
Co.  charge  him  $  500  for  the  piano.     How  much  do  they  make 
on  that  sale  ? 


236  PERCENTAGE 

216.  Mrs.  Lee  buys  from  Ball  &  Co.  a  piano  listed  at  $  700. 
She  agrees  to  pay  for  it  $750,  in  monthly  payments  of  $15 
each.     After  making  7  payments  she  returns  the  piano.     Ball 
&  Co.  spend  $  5  for  repairs  and  sell  it  for  $  625.     What  is 
their  gain  on  it  ? 

SUGGESTION  TO  TEACHER.  Require  the  pupils  to  bring  to  class  similar 
problems,  describing  business  occurrences.  Let  them  find  out  facts  which 
will  enable  them  to  keep  the  conditions  of  their  problems  within  the  range 
of  probabilities. 

Find  gain  on  the  following  goods  bought  at  discounts  of  10, 
20,  and  5  off,  and  sold  at  an  advance  of  20%  on  list  price  : 

Items  List  price 

217.  1300  yd.  Carpet  @       $  .75 

218.  800  yd.  Drapery  Silk    @     $  3.50 

219.  100  prs.  Lace  Curtains  @     $6.75 

220.  On   a  bill  of  $  675,  what   is  the  difference   between 
discounts  of  40%  and  20%,  and  a  straight  discount  of  60%? 

221.  Mr.  Dow,  a  merchant  in  Kentucky,  goes  to  New  York 
twice  a  year  to  buy  goods.     He  had  been  getting  discounts  of 
30,  20.  and  5%  off  for  cash,  but  on  his  last  trip  he  found  that 
he  could  get  discounts  of  30,  40,  and  5%.     If  he  bought  goods 
to  the  amount  of  $  9000,  how  much  was  he  benefited  by  the 
change  of  discounts  ? 

222.  The  list  price  of  an  article  with  three  different  houses 
is  $400.     One  house  offers  discounts  of  20,  10,  and  5%  ;  the 
second  5,  10,  and  20%  ;  the  third  10,  5,  and  20%.     Which  is 
the  best  offer,  and  why  ? 

INTEREST 

223.  Money  paid  for  the  use  of  money  is  called  Interest. 
The  sum  on  which  interest  is  paid  is  called  the  Principal. 

If  $  250  is  loaned  at  6  %  for  1  yr.,  how  much  is  the  interest  ? 
What  sum  is  the  principal  ? 


INTEREST  237 

224.  When  you  know  the  interest  of  a  sum  of  money  for  1 
yr.,  how  may  you  find  the  interest  of  the  same  sum  at  the  same 
rate  for  2  yr.  ?     7  yr.  ?     31  yr.  ?     ^  of  a  year  ?     2  yr.  1  mo.  ? 

Find  interest : 

Prin.       Rate          Time  Prin.  Rate          Time 

225.  $    200   6%  4yr.  232.  $1800  5%  1  yr.  8  mo. 

226.  $    700   8%  21  yr.  233.  $    400  7%  2  yr.  9  mo. 

227.  $    300   7%  21  yr.  234.  $    600  4%  1  mo. 

228.  $    800   8%  2  yr.  6  mo.  235.  $    800  6%  7  mo. 

229.  $    900   5%  3yr.  4  mo.  236.  $1200  3%  11  mo. 

230.  $    600   4%  2yr.  1  mo.  237.  $1800  9%  5  mo. 

231.  $5000   6%  3yr.  2  mo.     238.    $1500   8%    10  mo. 
There  are  many  ways  of  calculating  interest,  all  depending 

upon  this  fact.     Principal  x  Rate  x  Time  =  Interest.     A  6% 
method  and  a  cancellation  method  are  given  in  this  book. 

/Six  Per  Cent  Method 

239.    From  the  first  equation  reason  out  the  equations  which 
follow  it.     Learn  them. 

At  6%  the  interest  of  $  1  for  12  mo.  =  $  .06 
the  interest  of  $  1  for    2  mo.  =  $  k01 
the  interest  of  $  1  for    1  mo.  =  $  .005 
the  interest  of  $  1  for    6   da.  =  $  .001 
the  interest  of  $  1  for    1   da.  =  $  .0001 
At  6  %  what  is  the  interest  of  $  1.00  for : 


yr. 

mo. 

da. 

yr. 

mo. 

da. 

240. 

1 

2 

6 

246. 

8 

10 

18 

241. 

2 

1 

6 

247. 

9 

3 

1 

242. 

3 

1 

12 

248. 

7 

5 

5 

243. 

4 

3 

6 

249. 

8 

9 

7 

244. 

5 

3 

12 

250. 

6 

11 

8 

245. 

7 

8 

12 

251. 

7 

7 

9 

238  PERCENTAGE 

252.  CLASS  EXERCISE. may  give  a  number  of  years, 

months,  and  days,  and  the  class  may  find  the  interest  on  $  1 
for  that  time  at  6%. 

253.  Find  the  interest  of  $  1  at  6%  for  2  yr.  6  mo.  24  da. 
and  then  find  the  interest  of  $  2  for  that  time.    $3.    $7.    $  25. 

254.  Find  the  interest  of  $  25.75  for  5  yr.  8  mo.  12  da.  at  6%. 

25.75 

.342      The  interest  of  $  1  for  5  yr.  8  ino.  12  da.,  at  6%,  is  $.342. 
5150  The  interest  of  $25.75  is  25.75  times  $.342.     In  practice  it  is 
10300    more  convenient  to  multiply  25.75  by  .342.      Of  course  the 
7725      result  is  the  same. 
$8.80650 

Find  the  interest  of  $  125.37  at  6%  for: 

yr.          mo.          da.  yr.         mo.         da. 

255.  346  260.         297 

256.  7           9         12  261.  3         10           8 

257.  8           2         18  262.  659 

258.  7           5         24  263.  5           5         10 

259.  6         11           7  264.  11           2         11 
Find  interest  at  6%  : 

265.  $  175.25  for  1  yr.  6  mo.    7  da. 

266.  $  210.60  for  3  yr.  2  mo.    9  da. 

267.  $  625.48  for  2  yr.  9  mo.    9  da. 

268.  $  330.27  for  5  yr.  7  mo.    9  da. 

269.  $   45.60  for  8  yr.  3  mo.  10  da. 

270.  $  910.75  for  1  yr.  6  mo.  13  da. 

271.  $  712.25  for  2  yr.  8  mo.  14  da. 

272.  $  861.60  for  3  yr.  8  mo.  14  da. 

273.  $  520.40  for  7  yr.  4  mo.  14  da. 


INTEREST  239 


274.  What  is  the  interest  of  $100  for  lyr.  at  6%?   At  3%  ? 

275.  What  is  the  ratio  of  the  interest  of  a  sum  of  money  for 
a  given  time  at  3%,  to  the  ratio  of  the  same  sum  of  money  for 
the  same  time  at  6%  ? 

276.  Find  the  interest  of  the  following  amounts  for  1  yr. 
8  mo.  24  da.,  first  at  6%  and  then  at  3%  : 

$276        $24.76        $13.25        $417         $625 

277.  Find  by  the  6%  method  the  interest  of  $318  at  5% 
for  2  yr.  7  mo.  12  da. 

SOLUTION.  The  interest  of  $  318  for  2  yr.  7  mo.  12  da.  at  6  %  is  $  49.926. 
At  1  %  the  interest  is  -  as  much  or  $  8.321.  At  5  %  the  interest  is  5  times 
as  much  as  at  1  %  or  §  41.605. 

Required  interest  : 

Prin.  Rate  Time 

278.  $    28.35         6%  1  yr.    6  mo.  16  da. 

279.  $    49.36  7%  2  yr.  9  mo.  12  da. 

280.  $    30.75  8%  6  yr.               15  da. 

281.  $  252.00  6%  4  yr.  7  mo.  27  da. 

282.  $  160.00  7%  8  mo.  26  da. 

283.  $    72.00  5%  6  yr.  8  mo.  13  da. 

284.  $    75.00  4%  8  yr.  3  mo. 

285.  $  112.00  5%  6  yr.  7  mo.  22  da. 

286.  $    46.75  3%  2  yr.  11  mo.  20  da. 

287.  If  you  borrowed  $  100  and  paid  it  back  at  the  end  of 
one  year,  with  the  interest  on  it  at  6%,  how  much  would  you 
pay? 

288.  The  sum  of  the  principal  and  interest  is  called  the 
Amount. 

What   amount  must  be  paid  back  when  $  200  is  borrowed  at 
6%  interest  and  kept  2  yr.  10  mo.  18  da.  ? 


240 


PERCENTAGE 


Prin. 

Kate 

289. 

$900 

6% 

290. 

$800 

3% 

291. 

$600 

4% 

292. 

$144 

5% 

293. 

$672 

7% 

294. 

$  145.36 

7% 

295. 

$816.35 

5% 

296. 

$696 

8% 

297. 

$  216.25 

4% 

298. 

$625 

7% 

299.    CLASS  EXERCISE. 


Find  the  amounts  of  the  following: 

Time 
7  yr.    4  mo.    6  da. 

3  yr.  5  mo.    7  da. 

11  yr.  2  mo. 

8  mo.  12  da. 

3  yr.  8  mo.  15  da. 

2  yr.  2  mo.    1  da. 

3  yr.  3  mo.    3  da. 
5  yr.  7  mo.  13  da. 

4  yr.  18  da. 
9  yr.  9  mo.    9  da. 

may  mention  a  sum  of  money, 
and  the  class  may  find  the  amount  of  it  for  any  length  of  time 
and  at  any  rate  which  he  may  decide. 

300.  Find  the  time  from  Jan.  1,  1898,  to  July  7,  1899. 

301.  Mr.  Monroe  borrowed  $  300  Jan.  1, 1897,  at  6%.    What 
was  the  interest  March  1,  1898  ?     Sept.  1,  1899  ? 

302.  On  Nov.  7,  1896,  what  amount  was  due  on  $600  bor- 
rowed May  1,  1892,  with  interest  at  6%  ? 

303.  Find  the  amount  of  $376.25  borrowed  July  1,  1883, 
and  paid  Nov.  13,  1887,  with  interest  at  6%. 

At  6%,  what  is  the  amount  of  $  700? 

304.  Borrowed  Sept.    7,  1898          Paid  April  19,  1899 

305.  Borrowed  June  15,  1895 

306.  Borrowed   Dec.  12,  1891 

307.  Borrowed  Aug.    6,  1880 

308.  Borrowed  Feb.  29,  1896 


Paid  Oct.  3,  1898 
Paid  May  15,  1894 
Paid  May  12,  1885 
Paid  June  30,  1897 


INTEREST  241 

Cancellation  Method 

309.  If  the  interest  of  a  sum  of  money  for  a  certain  time  is 
$  72,  what  will  be  the  interest  of  that  sum  for  J  of  that  time  ? 
f  of  that  time  ? 

310.  The  interest  of  a  certain  sum  of  money  for  a  certain 
time  is  $  96.    Find  by  cancellation  the  interest  of  that  sum  for 
£|  of  that  time,     f  1  of  it.     £  £  of  it. 

311.  Find  the  interest  of  $  400  for  2  mo.  20  da.  at  3%. 

SOLUTION.  The  interest  of  $  400  for  1  yr.  at  3  %  is  400  x  jfo.  The 
interest  of  the  same  sum  for  2  mo.  20  da.,  or  80  da.,  is  -gfe  as  much. 
Hence  the  interest  =  ^  of  $  12. 

4  2 

Canceling  we  have  »  x  -|-  x  *  =  |  =  $2.66| 

J0J5       jJpjJ       6 

m 

3 
Find  the  interest  of  the  following  by  the  cancellation  method : 


Prin. 

Rate 

Time 

312. 

$    276 

12% 

3  mo.    9  da. 

313. 

$    184.50 

4% 

5  mo.  27  da. 

314. 

$1200 

6% 

1  mo.  21  da. 

315. 

$1400 

5% 

3  mo.  15  da. 

316. 

$1800 

7% 

3  mo.  20  da. 

317. 

$    625 

8% 

90  da. 

318. 

$   800 

1% 

63  da. 

319. 

$   900 

8% 

100  da. 

320. 

$1100 

6% 

33  da. 

321. 

$2175 

3% 

93  da. 

322. 

$4150 

6% 

63  da. 

323.    Find  the  interest  of  $840  at  5%  for  1  yr.  3  mo. 

840  x  T^  x  |f.     When  no  days  are  given,  find  the  number  ol  months 
and  express  them  as  twelfths  of  a  year. 

HORN.    GRAM.    SCH.    AR.  16 


242  PERCENTAGE 

Find  interest  : 

Prin.  Rate  Time 

324.  $560  3%  lyr.  8  mo. 

325.  $218.64  6%  2  yr.  1  mo. 

326.  $175.25  8%  1  yr.  6  mo. 

327.  $165.36  4%  4  yr.  2  mo. 

328.  $500  5%  7  mo.  6  da. 

When  days  are  given,  reduce  the  whole  time  to  days  and  express  as 
360ths  of  a  year. 

Find  interest  of  : 

Prin.  Rate                           Time 

329.  $700  2%  1  yr.  1  mo.    6  da. 

330.  $750  6%  3  yr.  1  mo.  15  da. 

331.  $420  1%  2  yr.  1  mo.  10  da. 

332.  $800  9%  2  yr.  3  mo.    6  da. 

333.  $875  10%  1  yr.  8  mo.  20  da. 

334.  $630.25  7%  1  yr.  4  mo.  20  da. 

335.  There  are  some  special  rules  for  calculating  interest 
which   are   derived   from   the  principles  of  the   cancellation 
method  as, 

To  compute  interest  at  8  %  — 

Multiply  the  principal  by  the  number  of  days,  move  the  decimal 
point  of  the  product  two  places  to  the  left,  and  divide  the  result  by  4$- 
Find  interest  of  $200  for  53  da.  at  8%. 

By  cancellation  method  By  special  rule 

2  53 


45)1060(2.35+ 
90 

It  will  be  seen  that  dividing  by  45  gives  160 

the  same  result  as  multiplying  by  8  and  135 

dividing  by  360. 


225 


INTEREST  243 

336.  Give  the  reason  for   the  special  rule  for  computing 
interest  at  8%. 

337.  Show  how  the  following  rules   are  derived  from  the 
cancellation  method : 

(a)  To  compute  interest  at  5  %  — 

Multiply  the  principal  by  the  number  of  days,  move  the  deci- 
mal point  of  the  product  two  places  to  the  left,  and  divide  the 
result  by  72. 

(b)  To  compute  interest  at  6%  — 

Multiply  the  principal  by  the  number  of  days,  move  the  deci- 
mal, point  of  the  product  three  places  to  the  left,  and  divide  the 
result  by  6. 

338.  Give  a  similar  rule  for  computing  interest  at  9%.     At 
4%.     At  12%.     At  10%.     At  3%. 

Exact  Interest 

339.  Usually  360  da.  are  considered  one  year,  but   some- 
times calculations  of  interest  are  made,  in  which  a  year  is 
considered  as  365  da.     This  is  called  Exact  Interest.     To  find 
the  exact  interest  of  a  sum  of  money,  use  the  cancellation 
method,  expressing  the  exact  number  of  days  as  365ths  of 
a  year. 

Find  the  exact  interest  of  $900  from  Dec.  1, 1898,  to  Feb.  12, 
1899,  at  8%. 

SOLUTION.  The  exact  number  of  days  from  Dec.  1,  1898,  to  Feb.  12, 
1899,  is  73  da.,  or  ^  of  a  year. 


Find  exact  interest  of  : 

340.  $300,  1  yr.  1  mo.  1  da.  at  9%. 

341.  $240,  11  mo.  25  da.  at  4%. 

342.  $336,  1  yr.  2  mo.  10  da.  at  6%. 

343.  $430  from  Oct.  15,  1897,  to  Jan.  11,  1898,  at 


244  PERCENTAGE 

PROMISSORY  NOTES 

344.  July  1,  1900,  Mr.  James  Allen  bought  a  horse  of 
Mr.  William  Brown  for  $125,  paying  $25  cash  and  giving 
a  promissory  note,  like  the  following,  for  the  balance.  The 
note  was  paid  in  full  when  due.  What  amount  was  paid  ? 


ilntfi,    neHUbi  nT,  A-i/x,  njpn.  n?/nl, 


jpn,  n/n/ruj/m, 


REVENUE 
STAMP 


O.HW. 


345.  A  Promissory  Note  is  a  written  promise  to  pay  money. 
Who  is  the  maker  of  the  above  note? 

346.  Who  should  keep  the  note  until  it  is  paid?     What 
should  be  done  with  it  after  the  money  is  paid  ? 

347.  Write  a  note  promising  to  pay  Robert  Ruskin  $300 
with  interest  at  6%  in  one  year  from  date. 

348.  The  person  to  whom  the  note  is  to  be  paid  is  called 
the  Payee.     Who   is   the   payee   of    the   note  you  have   just 
written  ? 

349.  The  sum  mentioned  in  the  note  is  called  the  Face  of 
the  note.     What  is  the  face  of  your  note  ? 

350.  The  date  at  which  the  note  becomes  due  is  called  the 
date  of  Maturity.     What  is  the  date  of  maturity  of  your  note  ? 

351.  Is  it  an  interest-bearing  note  ? 

SUGGESTION  TO  TEACHER.  Procure  blank  forms  of  promissory  notes 
of  different  kinds,  and  let  the  differences  between  them  be  discussed  in 
class. 

352.  What  is  the  amount  of  a  note  for  $425  that  matures 
in  4  mo.,  interest  being  5%  ? 


PROMISSORY  NOTES  245 

353.  In  some  states  the  law  allows  3  days  more  than  the 
specified   time   for   the  payment   of   a   note,   but   interest   is 
exacted  for  these  3  days,  called  Days  of  Grace.     If  a  note  is 
made  payable  Aug.  1,  on  what  day  is  it  really  due,  when  grace 
is  allowed  ? 

In  the  problems  of  this  book,  days  of  grace  are  not  to  be  considered 
unless  mentioned. 

354.  A  note  for.   $  400  is  dated  March  1,  1896,  and  made 
payable  May  1,  1896,  with  grace.     It  is  said  to  mature  May  1/4, 
and  interest  is  computed  to  May  4.     What   is  the  interest 
at  6%  ? 

355.  A  note  for  $  500,  dated  July  1,  1900,  is  made  payable 
in  3  mo.  with  grace.     When  does  it  mature,  and  what  is  the 
interest  at  6%  ?     At  5%  ?     At  7%  ? 

356.  How  much  must  be  paid  for  the  use  of  $  625  from 
June  1,  1897,  to  July  1,  1898,  with  grace,  at  6%  ?.    At  3%  ? 

357.  How  much  must  be  paid  for  the  use  of  f  500  from 
Dec.  1,  1899,  to  March  1,  1900,  with  grace,  at  6%  ?     At  4%  ? 

358.  What  is  the  interest  on  a  note  for  $  300  at  6%,  dated 
Aug.  31,  1895,  and  made  payable  in  30  da.,  with  grace  ? 

359.  June  17,  1897,  Mr.  Kent  gave  a  note  for  $500  at  6%, 
payable  in  60  da.,  with  grace.      When  was  the   note  due? 
What  was  its  face  ?     Its  amount  at  maturity  ? 

360.  Notes,  being  promises,  may  be  varied  to  suit  the  inten- 
tions of  the  parties  concerned.     Some  notes  draw  interest  from 
date,  some  after  maturity,  and  some  not  at  all.     Some  are  made 
payable  at  a  specified  time,  and  are  called  Time  Notes.     Some 
are  made  payable  upon  the  demand  of  the  holder  for  payment, 
and  are  called  Demand  Notes.     Some  are  made  in  such  a  way 
that  they  can  be  sold  or  transferred  to  other  persons,  and  are 
called  Negotiable  Notes.      Some  are  made  payable  only  to  a 
certain  person,  and  are  called  Non-negotiable  notes. 

What  is  the  special  advantage  of  a  negotiable  note?     Of 
a  non-negotiable  note  ? 


246  PERCENTAGE 

Demand  Note 

$300.75.  BOSTON,  MASS.,  Sept.  20,  1900. 

On  demand,  I  promise  to  pay  William  D.  Owen  three 
hundred  and  -^-fa  dollars,  with  interest  at  6  %.  Value  received. 

EDWARD  M.  ARLINGTON. 

361.  What  is  due  Jan.  19,  1901  ? 

362.  If  the  note  above  were  not  paid  until  May  20,  1902, 
how  much  would  be  due? 

363.  How  much  would  be  due  on  the  above  note  if  it  were 
paid  Sept.  25,  1902  ?     January  11,  1903  ?     March  1,  1903  ? 

364.  How  does  a  demand  note  differ  from  a  time  note? 

365.  Write  a  time  note  for  $  500,  due  in  3  mo.,  at  4%. 

Negotiable  Note 

$175.50.  LOWELL,  MASS.,  Sept.  15,  1899. 

One  year  after  date,  I  promise  to  pay  to  Henry  Scott,  or 
bearer,  one  hundred  and  seventy-five  and  -ffa  dollars,  with 
interest  at  6%.  Value  received.  MARY  GREEN 

366.  What  two  words  in  the  above  note  make  it  a  negotiable 
note? 

367.  If  the  negotiable  note  given  above  was  paid  when  it 
was  6  mo.  past  due,  how  much  was  paid  ? 

368.  Write  a  negotiable  time  note  for  $600,  interest  6%, 
and  find  the  amount  of  it  when  due.     When  3  mo.  past  due. 
11  mo.  past  due. 

$  1000.  SAN  FRANCISCO,  CAL.,  Nov.  12,  1901. 

One  year  after  date,  I  promise  to  pay  to  the  order  of  Ellen 
Eames,  One  Thousand  Dollars,  with  interest. 

JAMES  PORTER. 

369.  When  a  note  includes  the  words  "with  interest/'  but 
gives  no  specified  rate,  interest  is  computed  at  the  rate  legal  in 


PARTIAL   PAYMENTS  247 

the  state  in  which  it  is  dated.  Copy  the  above  note,  dating  it 
at  the  place  where  you  live,  and  find  the  amount  of  it  at 
maturity  under  the  laws  of  your  state. 

370.  Write  a  note  for  $  700  due  in  3  mo.,  with  interest  after 
maturity.      Find  the  amount  due  on  it  8  mo.  after  its  date 
under  the  laws  of  your  state. 

PARTIAL  PAYMENTS 

371.  May  7,  1896,  Mr.  James  Smith  gave  Mr.  John  Brown 
a  note  for  $  700  payable  on  demand,  with  interest  at  6%.    How 
much  was  due  May  7,  1897  ?     At  that  time  Mr.  Smith  made  a 
partial  payment  of  the  amount  due  by  giving  Mr.  Brown  $  442. 
On  how  much   money  ought  Mr.  Smith  to  continue  to  pay 
interest?     How  much  was  due  May  7,  1898?     At  that  time 
Mr.  Smith  made  another  partial  payment,  giving  $  218.     How 
much  was  due  on  that  note  Nov.  7,  1898  ? 

SOLUTION 

May  7,  '96,  Mr.  Smith  owed  Mr.  Brown $  700  Prin. 

Int.  on  prin.  from  May  7,  '96,  to  May  7,  '97  .     .     .     .     42  Int. 

May  7, '97,  Mr.  Smith  owed  Mr.  Brown $742  Am't. 

May  7,  '97,  Mr.  Smith  paid  Mr.  Brown 442  Pay't. 

Mr.  Smith  still  owed  Mr.  Brown $  300  New  prin. 

Int.  on  new  prin.  from  May  7,  '97,  to  May  7,  '98   .     .      18  Int. 

May  7,  '98,  Mr.  Smith  owed  Mr.  Brown $318  Am't. 

May  7,  '98,  Mr.  Smith  paid  Mr.  Brown 218  Pay't. 

Mr.  Smith  still  owed  Mr.  Brown $  100  New  prin. 

Int.  on  last  prin.  to  Nov.  7,  '98 3  Int. 

Nov.  7,  '98,  Mr.  Smith  owed  Mr.  Brown $103  Am't  due. 

372.  When  a  partial  payment  is  made,  the  holder  of  the 
note  writes  upon  the  back  of  it  the  amount  of  money  paid 
and  the  date  of  payment.    The  writing  is  called  an  Indorsement, 
and  serves  as  a  receipt  for  the  amount  paid.     What  were  the 
indorsements  that  Mr.  Brown  wrote  ? 

373.  Aug.  1,  '93,  Mr.  John  Dow  gave  to  Mr.  Frank  Rand 
his  note  for  $800  at  6%.     Aug.  1,  '94,  he  paid  $218.     Feb. 
1,  '95,  he  paid  $  223.90.     How  much  did  he  owe  Aug.  1,  '95  ? 


248  PERCENTAGE 

SUGGESTION  TO  TEACHER.  Let  the  pupils  in  one  section  of  the  class 
enact  the  part  of  Mr.  Dow  in  writing  the  note,  and  those  of  another 
section  take  the  part  of  Mr.  Rand,  making  the  indorsements  upon  the 
notes  written  by  the  others.  Let  class  discuss  justice  of  the  settlements. 

$  800.  NEW  ORLEANS,  March  1,  1898. 

For  value  received,  60  da.  after  date,  I  promise  to  pay  to  the 
order  of  Amos  Butler,  Eight  Hundred  Dollars,  with  interest 
at  6%.  HOWARD  CURTIS. 

374.  On  the   back  of  this  note   these   indorsements  were 
written   by  Mr.  Butler :    Dec.  1,  1898,  $  300.    June  1,  1899, 
$  222.08.    How  much  was  due  March  1,  1900  ? 

375.  Make  a  problem  in  which  you  suppose  that  you  give 
a  note  for  $  900  due  in  3  yr.,  with  interest  at  8%. 

What  amount  would  you  owe  at  the  end  of  3  yr.  if  you  made  no  pay- 
ment before  that  time  ?  On  what  principal  would  the  yearly  interest  be 
reckoned  ? 

But  suppose  that  instead  of  waiting  until  the  end  of  the  three  years 
you  made  a  payment  of  $  12  at  the  end  of  the  first  year.  The  interest 
then  due  would  be  $  72. 

If,  now,  in  this  case,  as  in  the  previous  problems,  the  payment  $  12 
were  deducted  from  the  amount  $  972,  and  if  the  difference,  $  960,  were 
regarded  as  a  new  principal,  observe  that  simply  because  you  had  made 
a  payment  on  the  note,  you  would  be  charged  interest  on  a  greater  prin- 
cipal. Would  that  be  just  ? 

376.  To  prevent  injustice  in  such  cases,  the  Supreme  Court 
of  the  United  States  has  adopted  the  following  rule : 

UNITED  STATES  RULE.  When  the  payment  is  less  than  the  interest  due 
at  the  time  of  payment,  no  change  of  principal  shall  be  made  at  that  time, 
but  the  interest  shall  be  computed  upon  the  same  principal  until  the  sum 
of  the  payments  shall  equal  or  exceed  the  interest  due. 

Make  a  problem  in  which  the  first  payment  on  a  note  is  less 
than  the  interest  due  when  the  payment  is  made. 

$  900.  CINCINNATI,  OHIO,  Sept.  30,  1896. 

One  year  from  date,  I  promise  to  pay  Henry  Moore,  or  order, 
Nine  Hundred  Dollars,  with  interest  at  8%.  Value  received. 

MARTIN  CAMPBELL. 

Indorsements  :   March  30,  1897,  '9  16;  Sept.  30,  1897,  $  56. 


PARTIAL   PAYMENTS  249 

377.  How  much  was  due  June  1,  1898? 

SOLUTION.  The  first  payment,  $  10,  is  less  than  the  interest,  $36,  that 
has  accrued  at  the  time  this  payment  is  made  (int.  of  $  900  for  6  mo.  at 
8%  =  $36).  Therefore,  we  compute  interest  to  the  time  of  the  second 
payment.  The  interest  of  $900  for  1  year  at  8%  is  $  72,  and  the  amount 
is  $972.  Subtracting  from  this  amount  the  sum  of  the  payments 
($16  +  $56  =  $72),  we  find  that  the  new  principal  on  Sept.  30,  1897, 
is  $900.  The  interest  on  $900  from  Sept.  30,  1897,  to  June  1,  1898 
(9  months),  is  $  54.  Therefore,  the  amount  due  June  1,  1898,  is  $ 954. 

378.  A  note  of  $280  was  dated  June  25,  '94,  interest  6%, 
indorsed  $  20,  Jan.  25,  '95.     How  much  was  due  June  25,  '95  ? 

379.  Face  of  note,  $  700.    Date,  July  2, '95.    Kate,  8%.     In- 
dorsed, Jan.  2,  '96,  $  225.     Find  the  amount  due  Oct.  2,  '96. 

380.  Face  of  note,  $200.     Date,  Sept.  7,  1896.     Rate,  7%. 
Indorsed,  March  7,  1897,  $30.     June  7,  1897,  $40.     Sept.  7, 
1897,  $  60.     Find  the  amount  due  Dec.  7,  1897. 

381.  When  settlement  is  made  within  a  year,  the  following 
rule  is  generally  used : 

MERCANTILE  RULE.  Find  the  amount  of  the  principal  from  date  to 
time  of  settlement.  Find  the  amount  of  each  payment  from  its  date  to 
the  time  of  settlement.  Subtract  the  amounts  of  the  payments  from  the 
amount  of  the  principal. 

Find  by  this  rule  the  amount  due  at  the  end  of  a  year  on 
a  note  of  $500  with  interest  at  6%,  if  a  payment  of  $200 
is  made  4  months  before  settlement. 

382.  A  note  for  $1000  was  given  Feb.  7,  1898.     Kate,  6%. 
Settlement  was  made  63  da.  later.     A  payment  of  $200  was 
made  30  da.  before  settlement,  and  15  da.  before  settlement 
$  300  was  paid.     How  much  was  paid  at  settlement  ? 

SOLUTION.  The  amount  of  the  principal,  $  1000,  from  date  of  note  to 
time  of  settlement  (63  da.),  is  $1010.50.  The  amount  of  first  payment, 
$200  (30  da.),  is  $201  ;  the  amount  of  second  payment,  $300  (15  da.),  is 
$300.75.  Subtracting  the  amounts  of  the  payments,  $501.75,  from  the 
amount  of  the  principal,  $1010.50,  there  remains  to  be  paid  $508.75. 

Find  amount  paid  at  settlement  applying  Mercantile  Rule. 

383.  Face  of  note,  $60.     Date,  June  20,  '85.     Rate,  8%. 
Indorsed,  $  20,  July  6,  '85.     Settled,  Aug.  23,  '85. 


250  PERCENTAGE 

384.  Face  of  note,  $  80.     Date,  Nov.  5,  '91.     Rate,  7%.     In- 
dorsed, $  30,  Dec.  5,  '91.     Jan.  5,  '92,  $  25.     Settled,  Feb.  5,  '92. 

385.  Face  of  note,  $  120.    Date,  Aug.  9, '93.    Bate,  6%.     In- 
dorsed, Sept.  15,  '93,  $  48.    Oct.  1,  '93,  $  45.    Settled,  Oct.  9,  '93. 

BANK   DISCOUNT 

386.  Mr.  James  Gage  sold  a  carriage  to  Mr.  John  Lyman, 
price  $  600,  terms  $  100  cash,  and  the  balance  by  a  note  due 
in  6  mo.  without  interest.     As  Mr.  Gage  wished  to  use  the 
money  in  his  business,  he  took  the  $  500  note  immediately  to 
Mr.  Peter  Reed,  who  discounted  it  at  8  %  ;  that  is,  in  exchange 
for   the   note,  he   gave   Mr.    Gage  what   remained   after  the 
interest  of  the  $  500  at  8  %  for  6  mo.  had  been  deducted  from 
the  $  500.      How  much  did  Mr.  Gage  receive  for  the  note  ? 
At  the  end  of  the  6  mo.  to  whom  should  Mr.  Lyman  pay  the 
$  500  ?     How  much  did  Mr.  Reed  make  by  the  transaction  ? 

387.  When  a  note  is  discounted,  the  payee  indorses  it,  mak- 
ing it  payable  to  the  one  who  discounts  it.      The  payee  is 
then  responsible  with  the  maker  of  the  note  for  its  payment. 
Mr.  Gage  wrote  on  the  back  of  the  note  when  he  transferred  it 
to  Mr.  Reed, 

Pay  to  the  order  of  Peter  Reed. 

JAMES  GAGE. 

If  when  the  note  became  due  Mr.  Reed  should  be  unable 
to  collect  the  amount  of  it  from  Mr.  Lyman,  to  whom  could 
he  look  for  payment  ? 

SUGGESTION  TO  TEACHER.  Let  three  pupils  enact  the  parts  of  Mr.  Gage, 
Mr.  Lyman,  and  Mr.  Reed,  one  making,  signing,  and  giving  the  note ; 
another  receiving,  indorsing,  and  transferring  it ;  the  third  discounting  it. 
Let  the  class  discuss  the  purpose  and  the  justice  of  each  step  in  the 
transaction. 

388.  To  discount  a  note  is  to  take  from  its  face  the  simple 
interest  on  it  for  the  time  between  the  date  of  discounting  and 
the  date  of  maturity. 


BANK   DISCOUNT  251 

At  5%,  what  is  the  discount  on  a  non-interest-bearing  note 
for  $  700  due  in  60  da.  ? 

389.  Discount  that  is  found   by  computing  interest  for  a 
certain  time  is  called  Bank  Discount. 

How  does  it  differ  from  trade  discount  ? 

390.  If  you  had  a  note  which  promised  to  pay  you  $  300  at 
the  end  of  a  year's  time  without  interest,  would  it  be  worth 
$300  now?     If  it  were  discounted  at  8%,  how  much  would 
the  discount  be,  and  how  much  would  you  receive  for  the  note 
now? 

391.  The  difference  between  the  bank  discount  and  the  face 
of  the  note  discounted  is  called  the  Proceeds  of  the  note. 

What  are  the  proceeds  of   a  non-interest-bearing  note  for 
$400  due  in  6  mo.  discounted  at  10%? 

392.  Hale  &  Co.  sold  a  carriage  for  $  400  ;  terms  $  50  cash, 
balance  by  note,  payable  in  60  da.,  without  interest.     As  they 
wished  to  use  the  money  at  once,  they  sent  the  note  to  a  bank 
where  it  was  discounted  at  8%.     What  were  the  proceeds  of 
the  note  ?     How  much  did  Hale  &  Co.  really  receive  for  the 
carriage  ? 

Find    bank   discount   and   proceeds  of  non-interest-bearing 
notes  for  the  following  amounts: 

393.  $  250,  due  in  90  da.,  discounted  at  6%. 

394.  $450,  due  in  30  da.,  discounted  at  9% 

395.  $900,  due  in  60  da.,  discounted  at  8%. 

396.  $  750,  due  in  60  da.,  discounted  at  9%. 

397.  $900,  due  in  30  da.,  discounted  at  6%. 

398.  $  650,  due  in  100  da.,  discounted  at  7  %  - 

399.    A  note  for  $800  due  in  4  mo.  was  discounted  at  6% 
3  mo.  before  it  was  due.     What  were  the  proceeds  ? 

As  the  note  had  only  3  mo.  more  to  run,  it  was  discounted  for  3  mo. 


252  PERCENTAGE 

Find  proceeds  of  non-interest-bearing  notes  discounted  at  8%  • 


400. 

Am't 
$600 

Date  of  note 

Feb.    1,  '97 

Date  of  discount 
Mar.    1,  '97 

Date  of  maturity 
Apr.  1,  '97 

401. 

$500 

Feb. 

15, 

'96 

Mar. 

31, 

'96 

May  1, 

'96 

402. 

$870 

Sept. 

1, 

'95 

Oct. 

10, 

'95 

Dec.  1, 

'95 

403. 

$660 

June 

4, 

'97 

July 

I, 

'97 

Aug.  4, 

'97 

404. 

$745 

Apr. 

6, 

'84 

June 

3, 

'84 

July  9, 

'84 

405.  Find  proceeds  of  a  note  given  April  15,  1894,  due  in 
60  da.,  discounted  May  15,  1894,  at  7%. 

Find  proceeds  of  the  following  non-interest-bearing  notes 
discounted  at  8%  : 

Face  Date  of  note  Time  Date  of  discount 

406.  $   38    June  7,  '95    60  da.     July  1,  '95 

407.  $  900  May  10,  '84  90  da.  June  10,  '84 

408.  $  850  Sept.  7,  '90  60  da.  Sept.  21,  '90 

409.  $  750  Mar.  15,  '87  4  mo.  May  1,  '87 

410.  $  1500  Oct.  2,  '93  60  da.  Nov.  1,  '93 

411.  If  you  had  a  note  promising  to  pay  you  $  400  in  one 
year  with  interest  at  6%,  how  much  would  it  be  worth  at  the 
end  of  the  year  ?     If  that  amount  were  discounted  at  8%,  what 
would  the  proceeds  be  ? 

As  national  banks  do  not  usually  discount  long-time  notes,  if  you 
wished  to  obtain  the  money  in  advance  on  this  note  you  might  apply  to  a 
savings  bank  or  to  one  of  those  persons  who  deal  in  money  and  are  called 
capitalists,  money  lenders,  brokers,  or  loan  agents.  Their  rates  of 
discount  and  their  ways  of  computing  it  vary.  In  the  case  of  interest- 
bearing  notes,  the  discount  is  sometimes  reckoned  upon  the  face  of  the 
note,  sometimes  upon  the  amount  due  at  maturity,  and  sometimes  the 
face  of  the  note  is  discounted  at  a  rate  per  cent  equal  to  the  difference 
between  the  rate  of  interest  and  the  rate  of  discount. 

412.  A  note  for  $  1500  due  in  one  year  with  interest  at  7% 
was  taken  to  a  money  lender,  who   deducted  8%  from    the 


BANK  DISCOUNT  253 

face.  What  were  the  proceeds?  At  the  end  of  the  year, 
how  much  did  the  money  lender  receive  ?  How  much  did  he 
gain? 

413.  A  note  for  $  1500  due  in  one  year  with  interest  at  7% 
was  taken  to  another  money  lender,  who  calculated  the  amount 
due  at  maturity  and  discounted  that  amount  at  8%.     What 
were  the  proceeds  ? 

414.  A  note  for  $  1500  due  in  one  year  with  interest  at  7% 
was  taken  to  another  money  lender,  who  agreed  to  discount  it  at 
8%.     He  found  the  difference  between  the  rate  of  interest  and 
the  rate  of  discount,  1%,  and  took  1%  of  the  face.     What  were 
the  proceeds  ? 

Find  the  proceeds  of  the  following  notes,  discounted  at  8% 
by  the  method  explained  in  Ex.  414 : 

In  these  problems,  the  notes  are  supposed  to  be  discounted  on  the 
days  on  which  they  are  dated. 

Face  of  note  Time  Rate 

415.  $  750  2  mo.  5% 

416.  650  3  mo.  6% 

417.  1700  1  mo.  6% 

418.  1200  4  mo.  7% 

419.  1000  60  da.  6% 

420.  Mr.  Ashby  bought  an  automobile  for  $  1000  and  sold 
it  for  $  1250,  receiving  $  1000  cash  and  the  balance  in  a  note 
due  in  6  mo.,  with  interest  at  6%.     On  the  day  of  the  sale  the 
note  was  discounted  at  8  %  by  the  plan  given  in  Ex.  414.    How 
much  did  Mr.  Ashby  gain  by  the  sale  ? 

421.  Mr.  Day  sold  three  bicycles,  each  of  which  cost  him 
$75,  011  the  following  terms:    For  the  first  he  received  $50 
cash  and  a  note  for  $50  due  in  4  mo.,  interest  6%.     For  the 
second  he  received  $  75  cash  and  a  note  for  $  25  due  in  6  mo., 
interest  5%.     For  the  third  he  received  $  99  cash.     The  notes 
were  discounted  at  date  at  8%  by  the  method  used  in  Ex.  414. 
Compare  the  profits  on  the  three  sales. 


254  PERCENTAGE 

INSURANCE 

422.  Mr.  Adams  has  a  house  worth  $  7000.     He  has  made 
an  agreement  with  the  agent  of   an   insurance  company  by 
which,  if  the  house  is  destroyed  by  fire,  he  will  receive  from 
the  company  $  5000,  or,  if  it  is  injured  but  not  destroyed,  he 
will  receive  a  sum  in  proportion  to  the  damage  done.     For  this 
insurance  against  loss  by  fire,  he  pays  the  company  every  year 
1%  of  the  sum  for  which  the  house  is  insured.     How  much 
does  he  pay  for  insurance  ? 

423.  The  money  paid  for  insurance  is  called  a  Premium. 
Mr.  Green,  wishing  to  provide  for  his  wife  in  case  of  his 

death,  has  taken,  for  her  benefit,  an  insurance  of  $  5000  upon 
his  life.  The  company  has  agreed  to  pay  her  $  5000  upon 
proof  of  his  death.  For  this  he  pays  a  premium  of  $27.35 
a  year,  for  each  thousand  dollars.  If  he  pays  premiums  for 
25  yr.,  what  amount  will  he  pay  to  the  company  ? 

424.  Mr.  Strong,  wishing  to  provide  for  his  future,  has  taken 
out  what  is  called   an   endowment  policy.     This  agreement 
provides  that  if  he  is  alive  at  the  end  of  ten  years,  he  shall 
receive  $3682.25,  and  in  case  of  his  death  at  any  time  during 
the  ten  years,  his  heirs  shall  receive  $  2500.     For  this  he  pays 
an  annual  premium  of  $  347.47.     To  how  much  will  his  premi- 
ums amount  in  the  ten  years? 

425.  There  are  two  kinds  of  insurance.      Property  Insurance 
and  Personal  Insurance. 

To  which  kind  of  insurance  does  each  of  the  three  preceding 
problems  refer? 

There  are  many  kinds  of  property  insurance,  as  insurance  against  loss 
by  fire,  tornadoes,  shipwreck,  theft,  unpaid  debts,  etc. 

426.  Mr.  Campbell  has  a  house  worth  $1800.    If  it  were 
insured  for  f  of  its  value  at  1  %  each  year,  what  would  be  the 
annual  premium?      If  the  house  were  destroyed  by  fire,  how 
much  insurance  would  Mr.  Campbell  receive  ? 


INSURANCE  255 

427.  What  would  be  Mr.  Campbell's  annual  premium  if  his 
house  were  insured  for  j  of  the  value,  and  the  rate  of  insurance 
were  1J%?     How  much  would  the  premium  be  if  the  rate  were 
1^%,  and  the  insurance  covered  £  of  the  value  of  the  house? 

428.  A  stock  of  goods  invoiced  at  $  10,500  was  insured  for 
f  of  its  value  at  l-j-%.     How  much  premium  was  paid  ? 

429.  A  ship  worth  $  75,000  was  insured  for  f  of  its  value  at 
If  % .    The  cargo,  valued  at  $  7500,  was  insured  for  £  of  its  value 
at  24%.     Find  amount  of  premiums. 

430.  Insurance  companies  generally  insure  property  for  a 
period  of  years,  as  1,  3,  or  5  yr.,  charging  a  certain  number  of 
cents  on  each  hundred  dollars  insured.      They  also  charge  a 
certain  amount,  usually  $  1,  for  the  written  agreement  to  pay 
the  insurance  in  case  of  loss.     This  written  agreement  is  called 
a  Policy. 

An  insurance  company  insured  Mr.  Allen's  house,  worth 
$ 1600,  for  f  of  its  value,  for  a  period  of  3  yr.,  charging  $  1.30 
for  each  $100,  and  $1  for  the  policy.  How  much  did  the 
insurance  cost  him? 

SUGGESTION  TO  TEACHER.  Let  an  insurance  policy  be  brought  into 
the  schoolroom  to  be  discussed  and  examined  by  the  pupils.  After  solv- 
ing the  following  problems  let  pupils  compose  similar  ones. 

431.  Mr.  Stevens  takes  out  an  insurance  policy  of  $7000 
for  a  period  of  3  years.    The  3  yr.  rate  is  twice  the  annual  rate, 
which  is  $  .65  for  $  100.     Policy,  $1.     Find  cost  of  insuring. 

Find  cost  of  insuring  the  following  property  at  the  above 
rates ;  the  policy  costing  $  1  in  each  case. 

432.  Barn,  $  600.     Hay,  $  300.     f  value  insured,  1  yr. 

433.  House,  $  2700.    Stable,  $  600.    £  value  insured,  for  3  yr. 

434.  Stock  of  goods,  value  $6000,  |-  insured,  1  yr. 

435 .  House,  $  5000,  $  4000  worth  of  insurance  taken  for  3  yr. 

436.  House,  $  9500,  $  7000  worth  of  insurance  taken  for  3  yr. 


256  PERCENTAGE 

437.  Mr.  Eice  takes  $3000  worth  of  insurance  from  the 
Helena  Fire  Insurance  Co.  for  a  period  of  3  yr.  at  the  rates 
given   above.     Policy,   $1.     Six  months   later   his   house  is 
damaged  by  fire  to  the  extent  of  $  1000,  for  which  amount  he 
receives  a  check  from  the  company.     How  much  better  off 
is   Mr.    Eice   than  he   would  have    been    had  he   taken   no 
insurance  ? 

438.  Mr.  Wood  insured  his  house  five  times  successively 
for  3  yr.  periods  at  $1.25  a  hundred,  each  policy  costing  $1. 
During  the  first  period  the  house  was  insured  for  $  8000,  and 
during  the  next  period  for  $  7500.     He  continued  to  reduce  the 
amount  of  insurance  $  500  each  time  he  renewed  it.     The  house 
was  never  injured  by  fire.     How  much  did  he  pay  out  for  in- 
surance on  the   house   during  those   15  yr.  ?     What  did  he 
receive  in  return  for  his  payments? 

439.  Mr.  Charles  Olney  insures  his  furniture  for  $500  for  3 
yr.     The  annual  rate  is  $  .55  per  hundred.      The  rate  for  3  yr. 
is  twice  the  annual  rate.     What  is  the  premium  ? 

440.  John  Gibson  took  out  two  policies:     $3000  on  dwell- 
ing, and  $12oO  on  furniture.     Term,  5  yr.     Annual  rate,  40^. 
5   yr.    rate   three  times  the    annual   rates.      What  was  the 
amount  of  both  premiums  ? 

441.  The  trustees  of  Perry  Township  hold  a  policy  on  a 
school  building  for  $3250.00.     Term,  5  yr.     What  premiums 
have  they  paid,  the  rate  being  as  in  the  previous  problem  ? 

442.  West  &  Co.  insure  their  stock  against  wind  storms,  for 
3  yr.,  for  $20,000.      Eate,  40^  a  hundred  for  3  yr.      What  is 
the  amount  of  premiums? 

443.  George  Brown  takes  out  a  policy  for  3  yr. :  $3000  on 
his  dwelling  and  $500  on  his  furniture.     Eate,  $.90  for  3  yr. 
Policy  fee,  $1.     What  is  the  cost  of  his  insurance? 

444.  A  house  valued  at  $  3000  and  insured  for  f  its  value 
was  struck  by  lightning.     The  adjuster  for  the  insurance  com- 


INSURANCE  f  TJ 

pany  estimated  that  it  was  damaged  37£%  of  its  value,  and  paid 
that  per  cent  of  the  amount  insured.  How  much  did  the 
owner  of  the  house  receive  from  the  insurance  company  ? 

There  are  several  kinds  of  personal  insurance,  as  Life  Insurance,  Acci- 
dent Insurance,  Endowments,  Annuities,  etc. 

445.  Mr.  Blake  took  out  a  life  insurance  policy  of  $5000 
for  the  benefit  of  his  wife,  upon  which  he  paid  $33.30  per 
thousand,  yearly   premium.      He  lived   20   yr.      How   much 
more  was  paid  to  his  widow  than  he  had  paid  to  the  insurance 
company  ?     What  would  have  been  the  difference  between  the 
amount  paid  to  the  company  by  Mr.  Blake  and  the  amount 
received  from  it  by  Mrs.  Blake,  if  he  had  taken  an  insurance 
of  $  10,000  ? 

'  Successful  insurance  companies  take  the  small  savings  of  those  who  are 
not  able  to  invest  them  to  advantage,  and,  massing  them,  invest  them  profit- 
ably. The  insured  loses  the  interest  of  the  money  which  he  pays  to  the 
company,  but  receives  an  assurance  that  those  for  whose  benefit  he  is 
insured  will  receive  the  full  amount  in  case  of  his  death  at  any  time. 

How  much  more  or  less  is  received  from  the  insurance  com- 
pany than  is  paid  to  it  in  the  following  cases  ? 

Amount  insured        Yearly  premium  per  $  1000          Years 

446.  $7,000  $30.70  20 

447.  6,000  44.84  40 

448.  10,000  28.28  21 

449.  30,000  27.38  50 

450.  50,000  35.65  5 

451.  200,000  39.00  15 

452.  Mr.  Bland  takes  out  a  life  insurance  policy  for  $  4000, 
paying  $  24  per  thousand.     What  is  the  annual  premium  ?     If 
he  pays  it  for  50  years,  how  much  more  does  he  pay  than  his 
heirs  receive  ? 

453.  Mr.  Corlen  insured  his  life  for  $  13,000  paying  his  first 
premium  of  $  57.50  per  thousand  on  Jan.  1,  1901.     He  died 
Mar.  1,  1901.     If  the  company  paid  the  agent  a  commission  of 

HORN.    GRAM.    SCH.    AR.  —  17 


258  PERCENTAGE 

15%  on  the  first  premium  for  insuring  Mr.  Corlen's  life,  how 
much  did  it  lose  by  the  insurance  ? 

454.  Mr.  Hill  took  out  a  life  insurance  policy  for  $  25,000 
at  the  rate  of  $32.75  per  thousand.      He  died  three  months 
after  paying  the  first  premium.     How  much  more  did  his  heirs 
receive  than  he  paid  ? 

455.  Many  insurance  companies  divide  a  part  of  their  earn- 
ings among  those  who  are  paying  premiums,  giving  each  one 
a  certain  per  cent  on  the  premium  he  pays.     This  amount  is 
called  a  Dividend. 

Find  the  value  of  a  dividend  of  5%  on  a  premium  of  $  250. 

456.  For  25  yr.  Mr.  Field  paid  an  insurance  premium  on 
a  life  policy  for  $  7000  at  the  rate  of  $  27.50  per  thousand. 
Ten  per  cent  of  the  amount  of  the  premiums  was  returned  to 
him  in  dividends.     How  much  did  his  insurance  cost  him  ? 

457.  Mr.  A.,  who  is  a  traveling  salesman,  carries  an  accident 
policy.     When  he  had  paid  $  205.75  in  premiums,  he  was  acci- 
dentally injured  and  received  an  allowance  of  $25  per  week 
for  7  weeks.     How  much  more  or  less  did  he  receive  from  the 
company  than  he  had  paid  to  it  ? 

458.  At  25  yr.  of  age  Mr.  B.  took  out  an  endowment  policy 
by  which  he  will  receive  $  5000  when  he  is  45  yr.  old.     An- 
nual premium,  $  240.38.    How  much  more  will  he  receive  from 
the  company  than  he  pays  to  it  ?     How  can  the  company  afford 
to  do  business  in  that  way  ? 

SUGGESTION  TO  TEACHER.  Let  pupils  make  problems  under  various 
imaginary  conditions,  getting  facts  about  insurance  from  agents  or  circu- 
lars in  order  that  their  problems  may  approximate  to  the  actual. 

459.  Neil  &  Co.,  agents  for  the  Westchester  Insurance  Co., 
insured  the  following  risks  for  periods  of  3  yr.  at  $  1.30  per 
hundred.     Their  commission  was  15%  on  the  premiums,  and 
they  received  from  the  insured  a  policy  fee  of  $  1  in  each  case, 
which  they  retained.     How  much  did  Neil  &  Co.  earn  and  how 
much  did  they  send  to  the  insurance  company  ? 


TAXES  259 

Dwelling,  value  f  1,800,  f  value  taken 

Store,  value     15,000,  -|  value  taken 

Stock  of  goods,  value    17,000,  %  value  taken 

Opera  house,      value    75,000,  f  value  taken 

460.  Miss  Otis  bought  from  an  insurance  and  annuity  com- 
pany a  yearly  annuity  of  $  100,  paying  for  it  $  1382.50.     How 
much  more  or  less  would  she  receive  from  it  than  she  paid  for 
it  if  she  lives  20  yr.  ?     7  yr.  ? 

461.  Mrs.  Green  owns  a  house  from  which  she  receives  a 
monthly  rental  of  $  25.     The  insurance  on  it  for  the  year  1898 
was   $  8  and  the  taxes  were  $  36.75.     It  was  vacant  three 
months.     How  much  was  her  net  income  from  it? 

TAXES 

462.  It  is  necessary  for  all  governments  to  tax  the  people 
to  pay  public  expenses.     Taxes  upon  property  are  calculated 
at  a  certain  per  cent  of  the  assessed  value. 

"At  1£%  how  much  are  the  taxes  upon  a  piece  of  property 
worth  $  9000  ? 

SUGGESTION  TO  TEACHER.  Explain  the  duties  of  the  assessor  and  tax 
gatherer.  Procure  copies  of  a  part  of  an  assessor's  list  of  taxables  and 
let  pupils  compute  the  taxes. 

463.  How  much  are  the  taxes  upon  Mr.  Hudson's  property 
which  is  assessed  at  $  8000,  where  the  rate  of  taxation  is 


464.  A  fixed  sum  assessed,  without  regard  to  their  property, 
upon  male  citizens  who  are  at  least  21  yr.  of  age,  is  called  a 
Poll  Tax. 

If  Mr.  Hudson  lived  where  the  poll  tax  was  $1.50,  how 
much  would  all  his  taxes  be  ? 

465.  Mr.  Howe,  who  pays  a  poll  tax  of  $  2,  owns  property 
assessed  at  $  6000,  in  a  city  where  the  rate  of  taxation  is  $  .75 
per  $  100,  and  $  2500  in  another  city  where  the  rate  of  taxation 
is  2^%.     What  are  his  taxes  ? 


260  PERCENTAGE 

466.  What  are  the  taxes  of  Mr.  Hearn  who  owns  real  estate 
assessed  at  $  11,375,  and  other  property  valued  at  $  2500,  the 
rate  of  taxation  being  l-g-%,  and  his  poll  tax  $  1  ? 

467.  Property  is  considered  to  be  of  two  kinds :  Real  Estate, 
as  lands,  houses,  stores,  factories,  mines,  and  other  immovable 
property ;  and  Personal  Property,  such  as  money,  notes,  furni- 
ture, and  other  property  which  can  be  carried  from  place  to 
place. 

Name  a  piece  of  real  estate.     Name  different  kinds  of  per- 
sonal property. 

468.  Mrs.  Kent  owns  real  estate  assessed  at  $  5600,  and  per- 
sonal property  assessed  at  $  1000.     The  rate  of  taxation  is 
$  1.20  on  $  100.     What  is  the  amount  of  her  tax  bill  ? 

469.  A  penalty  of  a  certain  per  cent  of  the  amount  of  the 
tax  is  sometimes  enforced  if  the  taxes  become  delinquent ;  that 
is,  if  they  are  not  paid  at  the  time  required.     If  Mrs.  Kent 
allows  her  taxes  to  become  delinquent,  and  the  penalty  is  10%, 
what  will  be  the  amount  of  her  tax  bill  ? 

470.  In  a  certain  city  the  rate  of  taxation  is  $  1.35  per  $  100, 
and  the  poll   tax  is   $  1.      How  much   are  the  taxes  of  an 
adult  male  citizen  whose  real  estate  is  valued  by  the  assessor 
at  $  1540,  and  personal  property  at  $  300  ? 

471.  What  is  the  tax  of  an  adult  female  citizen  whose  real 
estate  is  valued  at  $  1500,  and  personal  property  at  $  650  ? 

472.  Mention   several  things  which  are  paid  for  with  the 
money  raised  by  taxation. 

473.  Some  property,  as  churches,  government  bonds,  and 
public  property  of  all  kinds,  is  exempt  from  taxation.     Why 
are  public  school  buildings  not  taxed  ? 

474.  A  certain  town  raised  $  21,845  by  taxation,  a  part  of 
which  was  the  assessment  of  721  polls  at  $  2  each.     How  much 
tax  was  raised  from  property  ? 


TAXES  261 

475.  Taxes  to  the  amount  of  $  24,704.35  were  raised  in  the 
town  of  Nalasco,  of  which  $  23,456.35  was  raised  from  property, 
and  the  rest  from  polls  at  $  2  each.     How  many  citizens  paid 
poll  taxes  in  the  town  ? 

476.  In  a  town  where  there  are  1236  polls  assessed  at  $  1.50 
each,  it  was  decided  to  raise  $  46,854  by  taxation.     How  much 
must  be  raised  from  the  property  ?     If  the  property  valuation 
of  the  whole  town  was  $  3,600,000,  how  much  must  each  dollar's 
worth  of  property  yield  ? 

What  would  be  the  tax  of  each  of  the  following  residents  of 
that  town  ? 

477.  Mr.  A.,  31  yr.  old,  whose  realty  is  $  7000,  personals 
$  1375. 

478.  Mr.  B.,  19  yr.  old,  whose  realty  is  $  938,  and  who  has 
$  4500  worth   of   personal   property,  of   which   $  3000  is  in 
government  bonds. 

479.  Mrs.  C.,  whose  realty  is  $  7500,  personal  property  $  635. 

480.  Mr.  D.,  43  yr.  old,  who  has  no  property. 

481.  Miss  E.,  who  has  no  real  estate  and  $500  worth  of 
personal  property. 

482.  The  money  which  defrays  the  public  expenses  of  cities, 
counties,  and  states  is  raised  by  direct  taxation  upon  property 
or  person.     Money  for  the  expenses  of  the  national  government 
is  raised  by  indirect  taxation,  of  which  there  are  two  kinds, 
Internal  Revenue  and  Duties  or  Customs. 

The  internal  revenue  is  mostly  derived  from  taxes  on  the 
manufacture  of  liquors  and  tobacco  products  and  from  the  sale 
of  stamps  which  the  government  requires  to  be  placed  upon 
certain  legal  documents  and  articles  sold. 

To  defray  the  expenses  of  the  war  with  Spain  in  1898,  a  law 
was  passed  by  Congress  requiring  among  other  provisions  that 
a  one-cent  stamp  should  be  affixed  to  every  telegraphic  message 
or  express  receipt,  a  two-cent  stamp  to  every  bank  check, 


UNIVERSITY 


262  PERCENTAGE 

sight  draft,  etc.,  two  cents  for  every  hundred  dollars,  or  frac- 
tional part  thereof,  named  in  the  face  of  time  drafts,  prom- 
issory notes,  etc.,  and  stamps  of  different  values  upon  patent 
medicines,  proprietary  articles,  insurance  policies,  contracts, 
leases,  etc. 

What  should  be  the  value  of  a  stamp  affixed  to  a  promissory 
note  for  $  2500  ?  For  9  275  ?  For  $  39.50  ? 

483.  A  drug  firm  sold  in  one  week  1216  bottles  of  patent 
medicines,  each  requiring  a  stamp  whose  value  is  f  of  a  cent, 
1172  packages  each  requiring  a  stamp  costing  1^  ^,  and  298 
packages  each  requiring  a  stamp  costing  21  ^.     The  firm  sent 
15   telegrams   and   137   express   packages.     Fifty-one   checks 
were  given  by  the  firm.     How  much  revenue  accrued  to  the 
government  from  the  sale  of  stamps  necessary  for  the  business 
of  that  firm  for  that  week  ? 

484.  The  taxes  levied  by  the  government  upon  imported 
goods  are  called  Duties  or  Customs.    All  goods  which  come  into 
the  country  must  be  brought  in  at  certain  places  called  Ports 
of  Entry.     At  these  places  the  government  maintains  custom 
houses,  with  officers  who  collect  the  duties. 

There  are  two  kinds  of  duties,  Specific  and  Ad  Valorem. 

A  duty  of  a  certain  per  cent  of  the  amount  at  which  the 
goods  were  invoiced  in  the  country  from  which  they  were  im- 
ported is  called  an  Ad  Valorem  Duty.  The  Latin  phrase  ad 
valorem  means  "  according  to  value." 

Find  the  ad  valorem  duty  of  100  yd.  silk  invoiced  at  $  .50 
per  yd.,  duty  55%. 

At  the  rates  given,  how  much  ad  valorem  duty  would  be 
paid  by  a  firm  of  importers  upon  the  following  goods  ? 

485.  50  yd.  of  silk,  invoiced  at  $  1.25  per  yd.,  duty  55%. 

486.  500  pieces  of  ribbon,  10  yd.  in  a  piece,  at  75^  per  yd., 
duty  40%. 

487.  50  yd.  of  lace,  at  $  2.25  per  yd.,  duty  60%. 


TAXES  263 

488.  At  20%,  what  is  the  duty  on  75  bales  of  wool,  400  Ib. 
each,  invoiced  at  25  ^  per  pound  ? 

489.  At  25%,  what  is  the  duty  on  500  boxes  of  raisins,  each 
containing  40  Ib.,  costing  6J  cents  per  pound  ? 

490.  A  duty  levied  upon  a  certain  quantity  of  goods,  with- 
out reference  to  their  value,  is  called  a  Specific  Duty. 

If  the  specific  duty  is  $  2.25  per  dozen  pairs,  how  much  is 
that  duty  on  600  pairs  of  gloves  invoiced  at  50  f  a  pair  ?  If 
they  were  invoiced  at  75^  a  pair,  what  would  be  the  specific 
duty  ? 

491.  Sometimes  both   specific   and   ad  valorem  duties  are 
levied  upon  the  same  article. 

What  is  the  duty  on  30  pieces  of  carpet,  25  yd.  each,  in- 
voiced at  $1.75  per  yard,  the  specific  duty  being  25^  per 
yard,  and  the  ad  valorem  duty  40%  ? 

492.  What  is  the  duty  upon  800  Ib.  of  cigars,  invoiced  at 
$5  per  pound,  which  pay  a  specific  duty  of  $  4.50  per  pound 
and  25  %  ad  valorem  ? 

493.  An  importation  of.  silks  from  France  was  invoiced  at 
9324  fr.    At  60%  ad  valorem,  how  much  is  the  duty  in  American 
currency,  $  1  being  considered  equal  to  5.18  fr.  ? 

494.  What   duty  is  paid  by  an  American  importer  upon 
600  doz.  pairs  of  gloves  invoiced  at  60  fr.  per  dozen,  if  there 
is  a  specific  duty  of  $2  per  dozen  pairs  and  an  ad  valorem 
duty  of  40%  ? 

495.  Persons  are  allowed  to  bring  from  abroad  a  limited 
amount  of  goods  for  their  own  use  without  paying  duties  upon 
them.     An  American  lady  brought  home  from  Europe  a  silk 
dress  pattern,  upon  which  the  duty  was  $  31.75 ;  ^  doz.  pairs 
of  kid  gloves,  upon  which  the  duty  was  $  2.25  per  dozen  pairs ; 
20  yd.  of  lace  worth  f  2  per  yard,  upon  which  the  duty  was 

ad  valorem;  and  12  yd.  Irish  linen  at  60^  per  yard,  ad 


264  PERCENTAGE 

valorem  duty  35  %.  As  these  goods  were  for  her  own  use,  they 
were  passed  in  duty  free.  How  much  less  did  the  goods  cost 
her  than  they  would  have  cost  had  the  duty  been  collected  ? 

496.  If  an  importer  buys  700  yd.  of  velvet  at  $1.50  per 
yard,  pays  an  ad  valorem  duty  of  60%,  and  sells  it  at  $  4  per 
yard,  how  much  does  he  gain  ? 

497.  How  much  is  gained  by  an  importer  who  buys  20  pieces 
of  matting,  40  yd.  in  a  piece,  at  $  .10  per  yard,  pays  a  duty  of 
25%,  pays  for  transportation  $60,  and  sells  the  matting  at 
35^  per  yard? 

498.  A  list  of  articles  upon  which  duties  must  be  paid,  with 
the  special  duty  upon  each,  is  called  a  Tariff.      Tariffs   are 
changed  from  time  to  time  by  acts  of  Congress. 

An  importer  brought  through  the  custom  house  $  80,000 
worth  of  cut  glass  when  the  duty  was  35%  ad  valorem.  He 
sold  |  of  it  at  a  profit  of  25%  upon  invoice  price  plus  the  duty. 
The  tariff  upon  glass  was  raised  after  his  purchase  to  60%.  He 
sold  the  other  half  of  his  stock  at  a  profit  of  25%  upon  invoice 
price  plus  the  new  duty.  How  much  more  did  he  gain  on  the 
last  half  of  his  stock  than  on  the  first  half  ? 

499.  A  New  York  firm  imported  goods  invoiced  at  $64,000, 
upon  which  there  was  a  duty  of  12^%  ad  valorem.     For  how 
much  must  these  goods  be  sold  to  give  a  profit  of  20%  ? 

500.  Soon  after  those  goods  were  bought,  the  duty  on  that 
class  of  goods  was  changed  to  25%  ad  valorem.     Another  firm 
imported  $  64,000  worth  of  the  same  kind  of  goods  under  the 
new  tariff,  and  sold  their  goods  at  a  profit  of  20%.     How  much 
did  they  receive  ?     If  the  first  firm  sold  their  goods  for  the 
same  amount  as  the  second  firm,  how  much  more  did  they  gain 
than  the  second  firm  gained  ? 

501.  Mr.  Gilman  imported  $  100,000  worth  of  goods,  the 
duty  upon  which  was  30%  ad  valorem.     If,  after  he  sold  J  of 
them  at  a  profit  of  20%,  this  class  of  goods  was  put  on  the 


MISCELLANEOUS  EXERCISES  265 

free  list,  for  how  much  could  his  competitors  in  business 
buy  an  amount  of  goods  equal  to  what  he  had  left  on  hand  ? 
If  he  sold  the  rest  of  his  goods  for  20%  more  than  that 
sum,  would  he  gain  or  lose  on  the  whole  transaction,  and 
how  much  ? 

502.    Make  a  problem  in  which  an  importer's  business  is 
injured  or  benefited  by  changes  in  the  tariff. 

MISCELLANEOUS  EXERCISES 

1.  Resolve  into  prime  factors  6750.     7920. 

2.  Find  the  g.  c.  d.  of  235  and  685. 

3.  Find  the  1.  c.  m.  of  8,  10,  12,  16,  18,  20. 

4.  Divide  .012261  by  2.01. 

5.  3V64  +  7V81=? 

6.  2A/64  +  4^/125  =  ? 

7.  If  1  qt.  of  nuts  costs  11^,  how  many  bushels  can  be 
bought  for  $  13.20  ? 

8.  Find  the  interest  of  $  1240  for  5  yr.  9  mo.  27  da.  at  3%. 

9.  A  wheel  of  a  bicycle  is  7  ft.  in  circumference.     How 
many  times  does  tjie  wheel  turn  in  going  10  rd.  1  yd.  ? 

10.  John  weighs  115  lb.,  and  his  cousin  weighs   110  Ib. 
John's  weight  is  what  per  cent  of  the  sum  of  their  weights? 

11.  Write  the  following  decimally  and  as  common  fractions 
in  their  lowest  terms:    13%.     18%.     22|%.     158%.     875%. 


12.  How  much  is  30%  of  40  minus  16|%  of  66  ? 

13.  Find  11%  of  242.     37^%  of  162.     62i%  of  642. 

14.  Find  12|%  of  123.     66f  %  of  93.     87J%  of  123. 

15.  Square:  f.    .3.    1.2.    2f    .06.     ?i. 


266  PERCENTAGE 

16.  Name   four   numbers   between   100   and   200   that  are 
perfect  squares  and  give  their  square  roots. 

17.  What  number  between  100  and  200  is  a  perfect  cube  ? 

18.  Every  prime  number  greater  than  10  must  end  with 
either  1,  3,  7,  or  9.     Give  the  reason. 

19.  The  arc  AB  is  24  in.  long.     BC  is  50%  longer  than  AB. 
CD  is  33£%  longer  than  BC.     DA  is  25%  shorter  than  DC. 

How  long  is  the  circumference  ?  Diam- 
eter ?  Radius  ?  Perimeter  of  the  sector 
DO  A?  BOG'!  DOC?  AOB? 

c  20.  A  circumference  which  is  65  in. 
long  is  divided  into  two  arcs,  the  smaller 
arc  of  which  is  13  in.  long.  The  smaller 
arc  is  what  per  cent  of  the  greater  ? 

21.  How  many  coins  an  inch  in  diam- 
eter could  be  placed  in  rows  touching  one  another  on  a  rectangle 
4  in.  by  3  in.  ?  Represent. 

22.  When  the  hour  hand  of  a  clock  is  at  3,  what  per  cent  of 
one  revolution  around  the  clock  face  has  it  made  since  12  ? 

23.  At  4  P.M.  the  time  past  noon  is  what  per  cent  of  the 
time  before  midnight  ? 

24.  The  time  past  noon  is  what  per  cent  of  the  time  to  mid- 
night at  2  P.M.?     8  P.M.?     1.30  P.M.? 

25.  Thomas  wished  to  add  the  fractions  ^,  £,  and  i.     He 
first  multiplied  each  term  of  each  fraction  by  the  product  of 
all  the  denominators  except  its  own.      How  were  the  three 
fractions   then   expressed?      He   then  added  these  fractions 
and  reduced  their  sum  to  its  lowest  terms.     Was  his  process 
correct?     Can  you  show  a  better  way  to  find  the  sum  of 
these  fractions  ? 

26.  Take  Ex.  25,  substituting  the  fractions,  £,  -f,  and  Jy. 


MISCELLANEOUS  EXERCISES  267 

Fill  blanks  and  solve : 

27.  A  house  worth dollars  was  insured  for of  its 

value,  at per  cent.     What  was  the  annual  premium  ? 

28.  Mr.   A.    held   a   life   insurance   policy    for   $  2000,   on 
which  he  paid  an  annual  premium  of  $  52.     He  was  insured 
March  1,  1890,  and  died  June  1,  1900.     How  much  more  did 
his  heirs  receive  than  he  had  paid  out  in  premiums  ? 

29.  A  room  36  ft.  long  and  24  ft.  wide  is  to  be  covered  with 
carpet  f  yd.  wide,  at  $  1.10  per  yard.     How  much  will  it  cost 
if  the  strips  run  lengthwise  of  the  room  and  each  strip  is 
turned  in  4  in.? 

30.  Advance  the  following  goods  15%  in  price:    Caps  at 
30  t,  coats  at  $  8,  shoes  at  f  1.25,  gloves  at  78  f,  ties  at  15  £ 

31.  A  fisherman  caught  herring  enough  to  fill  500  barrels. 
He  sold  35%  of  the  catch,  and  kept  the  rest  for  a  rise  in  price. 
How  many  barrels  of  herring  did  he  keep  ? 

32.  How  many  quarts  of  berries  at  12-^  a  quart  would  be 
required  to  pay  for  9  yd.  of  cloth  at  16£^  a  yard? 

33.  Two  men  traveled  from  the  same  point,  one  east,  45^ 
mi. ;  the  other  west,  92f  mi.     How  far  apart  were  they  ? 

34.  Two  men  started  from  the  same  point,  and  traveled  in 
opposite  directions.     One  man  traveled  at  the  rate  of  7^  mi. 
per  hour,  the  other  at  the  rate  of  6J  mi.  per  hour.     How  far 
apart  were  they  at  the  end  of  1  hr.?     Of  3  hr.?     Represent. 

35.  John's  uncle  showed  him  a  half  eagle  one  morning,  and 
promised  to  give  him  at  night  25%  of  all  of  it  that  was  not 
spent.     At  night  his  uncle  reported  that  100%  of  the  money 
had  been  spent,  but  he  gave  him  75  $  instead.     How  much 
more  or  less  would  John  have  received  if  his  uncle  had  spent 
only  50%  of  the  value  of  the  half  eagle  ?    40%  ? 

36.  Nine  is  how  much  greater  per  cent  of  144  than  of  288  ? 


268  PERCENTAGE 

37.    How  much  is  gained  on  each  tablet  bought  at  the  rate 
of  $  1  per  dozen,  and  sold  at  10  $  each  ? 


38.  Mr.  Hall  earned  $  125  in  one  month,  which  was  62 

of  his  earnings  the  next  month.     How  much  did  he  earn  in 
both  months  ? 

39.  An  apple  tree  bore  21  bu.  of  apples,  which  was  87  -J% 
of  what  the  tree  next  to  it  bore.     What  was  the  difference  in 
the  yield  of  the  two  trees  ? 

40.  An  automobile  started  from  New  York,  and  ran  60  mi. 
the  first  day.     On  the  next  day  its  speed  was  33  \°]0  greater 
than  on  the  first  day,  and  on  the  third  day  it  was  25%  greater 
than  on  the  second.     How  far  was  the  automobile  from  New 
York  at  the  end  of  the  third  day  ? 

41.  What  number  plus  1%  of  itself  equals  909  ?     2424  ? 

42.  Forty-five  is  50%  more  than  what  number?     50%  less 
than  what  number  ? 

43.  A  man  had  $  654  in  bank.     He  drew  out  33£%  of  it, 
and  afterward  drew  out  25%  of  the  remainder.     How  much 
had  he  left  in  bank  ? 

44.  A  man  sold  a  wagon  for  $  180,  and  gained  25%.     What 
was  the  cost  of  the  wagon  ? 

45.  A  man  sold  a  wagon  for  $180  and  lost  25%.     What 
was  the  cost  of  the  wagon  ? 

46.  Near  the  close  of  summer  the  price  of  goods  costing 
$1.10  per  yard  was  cut  to  95^  a  yard.     What  per  cent  was 
lost? 

47.  Fifty  yards  of  cloth  were  bought  for  $30.     For  what 
price  per  yard  must  they  be  sold  to  gain  25%  ? 

48.  A  house  valued  at  $8000  was  insured  for  f  of  its  value 
at  1J%.     What  was  the  premium  ? 

49.  On  the  day  before  Christmas  Mary  counted  at  a  cer- 
tain corner  37  ladies  who  were  carrying  packages,  and  13  who 


MISCELLANEOUS   EXERCISES 


269 


had  no  packages.  What  per  cent  of  the  ladies  that  she  counted 
had  no  packages? 

50.  Thirteen  children  were  transferred  from  a  class  of  42. 
What  per  cent  of  them  remained  ? 

51.  In  making  peach  marmalade,  Mrs.  Harland  boiled  4  Ib. 
of  peaches  and  3  Ib.  of  sugar  in  a  quart  of  water.     Each  pint 
of  water  weighed  a  pound.     If  1  pt.  of  the  water  evaporated  in 
cooking,  what  per  cent  of  the  marmalade  was  sugar  ?    Peaches  ? 

52.  How  wide  is  a  rectangle  20  cm.  long  and  equal  to  f  of  a 

square  decimeter  ?     Represent. 

53.  In   the   rhomboid  ABCD  the 
line    BC    represents   10  ft.     AB   is 
50%  longer  than  BC.     How  long  is 
the  perimeter  ?     What  per  cent  of 

FIG.  2.  the  perimeter  is  AD ?     DC? 

54.  AB  and  DC  are  parallel.     DC 
=  48  ft.     AB  =  25%  of  DC.     BC  = 
250%   of  AB.     AD  =  100%  of  BC. 
Find  the  perimeter  of  the  trapezoid. 

55.  A  farm  is  in  the  shape  of  a 
trapezoid.     The  shorter  parallel  side 

is  16  rd.  long.  The  longer  parallel  side  is  121%  longer.  One 
of  the  non-parallel  sides  is  10  rd.  and  the  other  is  20%  longer. 
Represent.  Find  the  cost  of  fencing  the  farm  at  75^  per  rod. 

56.  What  is  a  trapezoid?     How  does  it 
differ  from  a  rhomboid? 

57.  A  four-sided  plane  figure  which  has  no 
two  sides  parallel  is  called  a  Trapezium. 

Draw  a  trapezium. 

58.  How  long  is  the  perimeter  of  a  trape- 
zium of  which  the   side  AB  is   3J-  in.,  the 
side  BC  5£  in.,  the  side  CD  6J  in.,  and  DA 

FIG.  4.  7|in? 


FIG.  3. 


270  PERCENTAGE 

59.  A  garden  is  fenced  in  the  form  of  a  trapezium.     One 
side  is  4  rd.  3  yd.  2  ft.  8  in.  long,  another  side  is  5  rd.  1  ft. 
10  in.  long,  another  side  is  4  rd.  5  yd.  4  in.  long.     The  other 
side  is  6  rd.  2  yd.  6  in.  long.    How  long  is  the  fence  ? 

60.  How  long  is  the  perimeter  of  a  trapezium,  the  shortest 
side  of  which  is  12  in.  long,  the  next  side  2  in.  longer  than  the 
first,  the  next  side  3  in.  longer  than  the  second,  and  the  last 
4  in.  longer  than  the  third  ? 

B  61.  In  the  trapezium  ABCD,  AD  re- 
presents 8  ft.  DC  represents  50%  more 
than  AD.  CB  represents  33^%  more  than 
DC.  BA  represents  121%  more  than  CB. 
How  long  is  the  perimeter  ? 

62.    John    had   a  kite    frame   in   the 
shape  of  a  trapezium  having  two  short 
FlG  5  equal  sides  and  two   long   equal   sides. 

If  a  long  side  was  3  ft.  long  and  a  short 

side  33|%  as  long,  what  was  the  combined  length  of  the  sticks 
that  made  the  frame,  allowing  -J-  an  inch  for  lapping  the  sticks 
at  each  angle  ? 

63.  Plane  figures  bounded  by  four  straight  lines  are  called 
Quadrilaterals. 

You  have  learned  six  different  kinds  of  quadrilaterals.    Draw 
one  of  each  kind  and  write  its  name  upon  it. 

64.  How  long  is  the  perimeter  of  a  rhombus  whose  sides  are 
each  1.7  in.  ? 

65.  Find  the  perimeter  of  a  rhomboid  whose  long  sides  are 
each  9.9  in.  and  whose  short  sides  are  each  5  in.  less  than  a 
long  side. 

66.  How  long  is  the  perimeter  of  a  trapezoid  if  one  of  the 
parallel  sides  is  7.65  in.,  the  other  8.45  in.,  and  each  of  the 
non-parallel  sides  is  4.7  in.  ?     Kepresent. 

67.  Plane  figures  bounded  by  straight  lines  are  Polygons. 
Name  four  kinds  of  polygons. 


MISCELLANEOUS   EXERCISES  271 

68.  What  name  is  given  to  a  polygon  of  3  sides  ?     5  sides  ? 
6  sides?     8  sides?     10  sides? 

69.  Is  a  sector  a  polygon  ?     Explain. 

70.  Mr.  K.  bought  9  doz.  pencils  for  $  2.16.     He  sold  them 
at  $  .03  apiece.     What  per  cent  was  gained  ? 

71.  500  sacks  of  coffee  were  bought  for  $300.     At  what 
price  per  sack  must  they  be  sold  to  gain  10%  ? 

72.  350  bottles  of  ink  were  bought  for  $  21.     For  how  much 
per  bottle  must  they  be  sold  to  gain  66f  %  ? 

73.  280  penknives  cost  $  70.     For  how  much  apiece  must 
they  be  sold  to  gain  25%  ? 

74.  A  dealer  paid  $  24  for  300  slates.     In  selling  them  he 
gained  50  % .     What  was  the  selling  price  of  each  ? 

75.  Two  gross  of  handkerchiefs  were  bought  for  $28.80. 
At  what  price  apiece  must  they  be  sold  to  gain  30%  ? 

76.  Mr.  Fowler's  salary  was  $  1800  a  year.     Last  year  he 
paid  $  1175.50  for  household  expenses,  $  22.50  for  life  insur- 
ance, $  15.75  for  taxes,  $  178.25  for  clothing,  and  $  23.75  for 
incidentals.     What  per  cent  of  his  salary  did  he  save  ? 

77.  A  grocer  sold  630  heads  of  cabbage,  which  was  66|%  of 
what  he  had.     How  many  had  he  left  ? 

78.  It  cost  $  15  to  build  a  certain  fence,  and  $  10  to  paint 
it.     The  cost  of  painting  was  what  per  cent  of  the  whole  cost  ? 

79.  A  quart  of  water  was  added  to  6  gallons  <ff  cider.     What 
per  cent  of  the  mixture  was  water  ? 

80.  Eight  pounds  of  river  water,  when  distilled,  furnished 
7f  Ib.  of  pure  water.    What  per  cent  was  removed  by  distilling  ? 

81.  In  making  brown  bread,  Mrs.  Goodwin  mixed  one  cup- 
ful of  white  flour,  one  cupful  of  Graham  flour,  and  four  cupfuls 
of  corn  meal.     What  per  cent  of  each  was  in  the  mixture  ? 


272  PERCENTAGE 

82.  Ten  years  ago  Mr.  D.  paid  $  4000  for  a  house  and  lot, 
which   has   increased    in  value   18%.      What   is   its   present 
value  ? 

83.  By  the  sale  of  a  horse,  a  man  gained  $  10,  which  was 
12£%  of  what  he  gave  for  it.     For  how  much  did  he  sell  it? 

84.  A  house  worth  $4000  depreciates  in  value  to  $3280. 
What  per  cent  does  it  depreciate  ? 

85.  If  I  buy  2000  bu.  of  wheat  at  85  cents  a  bushel,  and  sell 
it  for  $  340  more  than  I  paid  for  it,  what  per  cent  shall  I  gain  ? 

86.  A  boy  buys  papers  for  2  cents  each,  and  sells  them  at 
a  gain  of  150%.     What  price  does  he  get  for  them  ? 

87.  A  book  agent  receives  a  commission  of  28%  on  all  his 
sales.     If  he  gets  orders  for  125  books  at  $  2.50  each,  how 
much  does  he  gain  ? 

88.  If  I  send  orders  amounting  to  $  28.75,  getting  a  discount 
of  25%  with  5%  off  for  cash,  how  much  money  must  I  send  ? 

89.  About  80%  of  a  human  body  is  water.     At  that  rate,  if 
a  man  weighs  165  lb.,  how  much  of  his  body  is  water? 

90.  What  per  cent  of  75  is  each  of  the  first  12  multiples 
of  6}? 

91.  What  per  cent  is  gained  or  lost  by  buying  goods  at 
$  .33£  per  yard  and  selling  them  at  $  .25  ?     $  .50  ?     $  .66|  ? 


92.  What  per  cent  is  gained  by  buying  apples  at  the  rate 

of  3  for  a  cent  and  selling  them  at  the  rate  of  2  for  a  cent  ? 
• 

93.  Mr.  Chapman  built  a  house  which  cost  $  3600.     He  had 

$  3250,  and  borrowed  the  rest  June  1,  1895,  giving  his  note  at 
6%.  June  1,  1896,  he  paid  the  interest  due  and  $  100  of  the 
principal.  June  1,  1897,  he  paid  the  interest  and  $  50  of  the 
principal.  How  much  was  due  June  1,  1898  ? 

94.  Mr.  S.  built  a  house  for  Mr.  M.,  on  which  he  made  a 
profit  of  25%.     He  received  in  payment  $2000  and  two  lots 


MISCELLANEOUS   EXERCISES  273 

at  $  500  each.  The  lots  depreciated  in  value  $  50  each  before 
he  sold  them.  How  much  was  his  real  profit  on  the  house  ? 
What  per  cent  ? 

95.  A  wholesale  dealer  pays  $  260  for  a  car  load  of  bananas 
containing  520  bunches,  and  sells  them  at  $  .60  a  bunch.   What 
per  cent  does  he  gain  ? 

96.  A  retail  dealer  pays  60  f  for  a  bunch  of  bananas  contain- 
ing 8  doz.     If  he  sells  them  at  15^  a  doz.,  what  per  cent  does 
he  gain  ?     What  per  cent  does  he  gain  on  the  whole,  if  \  of 
them  spoil  before  he  sells  them  ? 

97.  Mr.  Malone,  a  traveling  salesman,  sold  Mr.  J.  W.  Smith 
of   Salem,  111.,  $360  worth  of   staples,  and  $170  worth  of 
notions.     He  was  allowed  a  commission  of  2%  on  the  staples, 
and  6%    on  the   notions.      How    much   commission  did  he 
receive  ? 

98.  Mr.  Perry  brings  49  bu.  of  wheat  to  the  Melrose  Mill  to 
exchange  for  flour.     If  he  gets  36  Ib.  of  flour  for  every  bushel 
of  wheat,  how  many  sacks  of  flour,  each  weighing  98  Ib.,  will 
he  get  ? 

99.  What  number  increased  by  62  J-%  of  itself  equals  3214  ? 

100.  Mr.  Taylor  bought  6  loads  of  hay,  each  weighing  1J  T., 
at  $  10  a  ton.     He  sold  all  of  it  for  $  99.     What  was  the  gain 
per  cent  ? 

101.  A  farmer  had  a  field  21  rd.  square.     Three  rows  of 
wire  fencing  were  put  around  it,  costing  2^  a  foot,  5%  off  for 
cash.     If  he  paid  cash,  how  much  did  the  fence  cost  ? 

102.  33^-%  of  the  fence  was  blown  down.    The  price  of  wire 
having  risen  50%,  and  the  cash  discount  being  the  same,  how 
much  will  it  cost  to  replace  the  fence  if  he  pays  cash  ? 

103.  Mr.  Gibson's  salary  is  $  6000  a  year.     Last  year  he 
saved  20%  of  the  first  three  months'  salary,  40%  of  the  next 
three  months'  salary,  and  30%  of  the  last  six  months'  salary. 
How  much  did  he  save  during  the  year  ? 

HORN.    GRAM.    8CH.    AR.  — 18 


274  PERCENTAGE 

104.  W.  H.  Small  &  Co.  bought  10  loads  of  hay,  weighing 
2700  Ib.  each,  at  $  10  per  ton,  and  sold  it  immediately  to  Mr. 
Knox  at  a  profit  of  20%.     Mr.  Knox  gave  in  payment  a  note 
payable  in  60  da.     If  it  was  discounted  in  bank  at  date  of  issue, 
at  7%,  how  much  did  Small  &  Co.  gain  by  the  transaction? 

105.  Arthur  and  Edward  bought  a  paper  route,  paying  $  10 
for  it.     Arthur  put  in  $  4,  and  Edward  the  remainder.     What 
fractional  part  of  the  route  belongs  to  Arthur  ?     To  what  per 
cent  of  the  profits  is  he  entitled  ?     In  one  week  the  earnings 
were  $  7.50.     How  much  is  each  boy's  share  ? 

106.  Mr.  A.  and  Mr.  B.  hire  a  pasture  for  $  50.     Mr.  A.  puts 
in  5  cows,  Mr.  B.  15  cows .     Mr.  A.'s  cows  are  what  per  cent 
of  the  whole  number  of  cows  ?     How  much  ought  he  to  pay 
as  his  share  of  the  cost  of  the  pasture  ? 

107.  Mr.  C.  owns  f  of  a  business,  the  profits  of  which  last 
year  were  $  16,000.     What  was  his  share  of  the  profits  ? 

108.  Mr.  Davis  owns  51%  of  a  business  which  year  before 
last  lacked  $  1000  of  paying  expenses.     How  much  money  was 
he  obliged  to  advance  in  order  to  keep  the  business  running  ? 

109.  Last  year  the  receipts  of  the  business  were  $  1248.75 
more  than  the  expenses.     How  much  did  Mr.  Davis  receive 
from  it  ? 

110.  If  10  hr.  are  considered  a  day's  work  and  $  2  a  day's 
pay,  what  are  the  weekly  earnings  of  a  man  who  works  9  hr. 
on  Monday,  8J  hr.  on  Tuesday,  11  hr.  on  Wednesday,  7  hr. 
on  Thursday,  6|-  hr.  on  Friday,  and  5  hr.  on  Saturday  ? 

111.  What  would  be  his  weekly  earnings  for  the  same  num- 
ber of  hours'  work  at  $  2.50  per  day  ? 

112.  What  would  be  his  weekly  earnings  for  the  same  num- 
ber of  hours'  work  at  $  2  a  day,  8  hr.  being  considered  a  day's 
work  ? 

113.  At  $  2  a  day  for  an  8-hour  day,  calculate  the  weekly 
earnings  of  each  man  in  the  following  time  sheet : 


MISCELLANEOUS  EXERCISES  275 


Mon. 

Tues. 

Wed. 

Thurs. 

Fri. 

Sa 

a 

Mr. 

Cox, 

7 

*i 

7} 

6| 

7 

7: 

b 

Mr. 

Dow, 

8 

8i 

6 

7i 

7i 

6 

c 

Mr. 

Lee, 

n 

8| 

9 

9 

6i 

5 

d 

Mr. 

Van, 

5£ 

6i 

8 

8} 

7i 

6 

e 

Mr. 

Gay, 

7 

6 

7* 

8 

8 

7 

114.  Julia  saved  money  to  buy  a  typewriter,  the  price  of 
which  was  $80.     When  she  had  saved  $48,  what  per  cent 
of  the  price  was  lacking  ?     When  she  had  paid  $  16  more, 
what  per  cent  of  the  price  was  still  unpaid  ? 

115.  Three  boys  bought  a  wagon,  John  paying  $  1,  James 
$  2,  and  Charles  $  3.     They  sold  the  wagon  for  $  7.50.     How 
much  was  each  boy's  share  of  the  gain  ? 

116.  A  man  received  a  legacy  of  $  5000.     He  bought  a 
house  and  lot  with  40%  of  it,  purchased  $  1500  worth  of  bank 
stock,  and  invested  the  remainder  in  business.     What  per  cent 
of  it  did  he  invest  in  business  ? 

117.  At  $  1.50  per  cord,  how  much  will  a  man  earn  by 
sawing  a  pile  of  wood,  16  ft.  long,  8  ft.  wide,  and  4  ft.  high  ? 

118.  Find  the  amount  of  the  following  bill : 

4  rms.  sandpaper  @  $  3.50,  discounts  30%  and  10%. 
3  doz.  packages  tacks  @  $  5.50,  discounts  50%  and  5%. 
12  car  jacks  @  $  6.00,  discounts  40%,  10%,  and  5%. 
2000  tin  rivets,  $  1.44,  discounts  40%  and  10%. 

119.  A  man  received  a  legacy.     After  investing  66f  %  of  it 
and  spending  25%  of  it,  he  had  left  $  1728.     How  much  was 
the  legacy  ? 

120.  A  man  having  $  4800  in  a  bank,  drew  out  12-J-%  of  it 
and  then  deposited  a  sum  which  was  75%  of  what  he  had 
drawn  out.     How  did  his  bank  account  stand  then  ? 


276  PERCENTAGE 

121.  Mr.  Boyd  sold  a  watch  for  $  45,  which  was  25%  more 
than  he  gave  for  it.     For  how  much  must  he  have  sold  it  to 
gain  66f  %  on  it  ? 

122.  Simplify: 
abed 

^  JQff  1*  iij: 

A  t«f*  1*  i  +  i 

123.  How  long  is  the  circumference  of  a  circle  whose  radius 
is  5$  in.  ? 

124.  A  burial  lot  in  the  form  of  a  circle,  40  ft.  in  diameter, 
is  inclosed  by  a  fence.     Find  the  cost  of  the  fence  at  $  .12J  a 
foot. 

125.  A  calf  is  tied  to  a  tree  by  a  rope  10  ft.  long.     The  rope 
can  slip  around  the  trunk  of  the  tree,  which  is  1  ft.  in  diameter. 
As  the  calf  runs  around  the  tree,  how  long  is  the  circumference 
of  the  largest  circle  he  can  make  ? 

126.  Reduce  5  qt.  1.725  pt.  to  decimals  of  a  bushel. 

127.  14  x  (— )  x  2.3  x  -f  =  3.91.     Find  the  missing  factor. 

128.  How  many  square  centimeters  are  there  on  the  surfaces 
of  7  1.  ? 

129.  In  a  park  there  is  a  circular  fountain  35  ft.  in  diameter, 
bordered  by  a  gravel  walk  7  ft.  wide.     What  is  the  distance 
around  the  outer  edge  of  the  walk  ? 

130.  A  swimming  pool  is  60  ft.  long,  20  ft.  wide,  and  the 
water  in  it  is  7  ft.  deep.     How  many  cubic  feet  of  water  are 
in  it? 

131.  Draw  a  trapezoid,  making  one  of  the  parallel  sides  6 
in.  long,  the  other  8  in.  long,  and  the  shortest  distance  between 
them  4  in.     Find  the  area  of  the  trapezoid. 

132.  A  farmer  has  a  field  in  the  shape  of  a  trapezoid.     One 
of  the  parallel  sides  is  90  rd.  long,  the  other  is  120  rd.  long, 
and  the  shortest  distance  between  them  is  30  rd.     How  many 
acres  in  the  field  ? 


MISCELLANEOUS  EXERCISES  277 

133.  One  of  the  sides  of  a  rhomboid  is  12  in.,  and  a  side 
adjacent  to  it  is  8  in.     Eepresent  and  tell  what  per  cent  of  the 
perimeter  each  side  is. 

134.  One  of  the  parallel  sides  of  a  trapezoid  is  10  in.,  which 
is  83|-%  of  the  other  parallel  side.     Each  of  the  non-parallel 
sides  is  9  in.    Represent  and  find  what  per  cent  of  the  perimeter 
each  side  is. 

135.  A  pupil  drew  a  square  and  shaded  11%  of  it.     If  the 
area  of  the  part  shaded  was  44  sq.  in.,  what  was  the  area  of  the 
whole  square  ? 

136.  What  per  cent  of  the  perimeter  is  each  side  of  a 
rhombus?     Of  a  regular  hexagon?     Of  a  regular   octagon? 
Of  a  regular  decagon? 

137.  A  jeweler  sold  a  watch  for  $  90,  gaining  20%.     What 
per  cent  would  he  have  gained  if  he  had  sold  it  for  $  100  ? 
$110?     $120? 

138.  Mr.  Baker  bought  $  100  worth  of  goods  and  sold  f  of 
them  for  what  ^  of  them  cost.     What  per  cent  did  he  gain  ? 

139.  What  per  cent  is  gained  or  lost  by  buying  $  100  worth 
of  goods  and  .selling  \  of  them  for  what  f  of  them  cost? 

140.  An  importer  buys  in  France  1000  Ib.  of  perfumery 
invoiced  at  $  2.00  per  pound.     Specific  duty  60^  per  pound,  ad 
valorem  duty  45%.     He  sells  it  at  $  5  per  pound.     How  much 
does  he  gain  and  how  much  does  the  government  gain  by  the 
transaction  ? 

141.  On  Aug.  1,  1890,  a  man  insured  his  life  to  the  amount 
of  $15,000  in  favor  of  his  wife,  paying  an  annual  premium  of 
$  29.30  per  $  1000.     He  died  Oct.  17,  1897.     How  much  more 
than  he  had  paid  did  his  widow  receive? 

142.  A  farmer  brought  to  market  24  doz.  eggs,  21  chickens, 
and  20  Ib.  of  butter.     He  sold  -f  of  the  chickens  at  25^  apiece, 
all  the  butter  at  15^  a  pound,  and  75%  of  the  eggs  at  10^  a 


278  PERCENTAGE 

dozen.  How  many  eggs  and  chickens  did  he  have  left  ?  He 
took  groceries  to  the  amount  of  |  of  his  sales.  He  bought  10 
yd.  of  calico  at  7^  a  yard,  and  a  pair  of  shoes  for  $  2.75.  How 
much  money  had  he  left? 

143.  A  boy  made  a  chicken  coop,  using  a  bundle  of  laths 
which  cost  15^,  and  5^  worth  of  nails.     He  sold  the  coop  for 
$  .75.     What  was  his  gain  per  cent? 

In  the  following  shipments  find  the  amount  belonging  to 
each  railroad  company : 

144.  From   Trent's  Landing,  Ky.,  to  Milwaukee,  Wis.,  via 
A.  &  B.,  B.  &  E.,  and  E.  &  M.:  Empty  kegs,  20,000  lb.;  rate, 
11^  per  hundredweight.     The  A.  &  B.  receive  25%   of  the 
freight  charges.     Of  the  remainder,  the  B.  &  E.  receive  45%, 
and  the  E.  and  M.  55  < 


145.  From  Mather,  111.,  to  Cincinnati,  Ohio,  via  M.  W.  &  E. 
and  A.  &  C. :  1  car  hogs,  18,000  lb. ;  rate,  20^^  per  hundred- 
weight.    The  M.  W.  &  E.  receive  25%,  and  the  A.  &  C.  the 
remainder. 

146.  From  Leeds,  Ky.,  to  Eawley,  Minn.,  via  L.  &  H.,  E.  & 
C.,  Chicago  &  St.  Paul,  St.  P.  &  R.:  25 %  tons  pig  iron;  rate, 
$2.75  per  ton.     15^  a  ton  goes  to  the  Ohio  Bridge  Co.;  the 
lines  south  of  Chicago  receive  55%  of  the  remainder.      The 
lines  north  of  Chicago  receive  the  balance.     The  L.  &  H.  re- 
ceives 25%  of  the  amount  paid  to  the  southern  lines,  and  the 
St.  P.  &  R.  25%  of  the  amount  paid  to  the  northern  lines. 

147.  Mr.   Ward  bought  a  lot  for  $600  and  built  on  it  a 
house  worth  $  2400.    The  property  was  assessed  at  f  its  value, 
and  the  tax  rate  was  1^-%.     The  insurance  was  for  f  of  the 
value  of  the  house,  and  the  rate  was  40  ^  a  hundred.    He  rented 
the  house  for  a  year  at  $  30  a  month.    How  much  more  did  he 
receive  the  first  year  from  his  investment  of  $  3000  than  he 
would  have  received  by  putting  it  at  interest  at  6%? 

148.  The  second  year  the  house  was  vacant  one  month  and 
required  $  10.65  worth  of  repairs.     Insurance  and  taxes  were 


MISCELLANEOUS  EXERCISES  279 

the  same  as  the  first  year.     How  much  more  were  the  net  re- 
ceipts than  6  %  of  the  amount  invested  ? 

149. '  The  third  year  the  rent  was  raised  to  $35  a  month. 
The  repairs  cost  $  48.75,  and  the  house  was  vacant  two  months. 
Compare  the  net  receipts  with  6%  on  the  investment. 

150.  Make  problems  showing  the  expenses  and  receipts  of 
Mr.  Ward's  house  for  successive  years. 

151.  At  the  end  of  7  yr.  Mr.  Ward  sold  the  house  and  lot 
for  $4500.     If  he  had  received   as   rental  during  the  7  yr. 
$2000  more  than  the  expenses  of  the  house  for  that  time, 
what  average  yearly  per  cent  had  he  gained  on  his  investment  ? 

152.  By  selling  a  house  for  $  1964,  20%  was  lost.     What 
would  have  been  the  selling  price  if  only  5%  had  been  lost  ? 

153.  Room  No.  5  in  Baker  School  has  44  desks  in  it.     If 
$  77  was  87%%  of  the  cost  of  all  of  them,  how  much  did  one 
desk  cost  ? 

154.  A  bought  640  A.  of  woodland  at  $38  per  acre.     He 
sold  the  timber  for  $  19,600,  and  the  land  for  $  13  per  acre. 
Find  the  per  cent  of  gain. 

155.  An  insurance  policy  for  $2200  cost  $17.60.     What 
was  the  rate  of  premium  ? 

156.  An  agent  for  an  oil  company  sells   in  three  weeks 
$48,000  worth  of  oil  at  a  commission  of  l-g-%.     If  his  expenses 
are  $  47.50  per  week,  how  much  are  his  net  earnings  for  that 
time? 

157.  62|%  of  200  =  how  many  times  121  ?     6J  ? 

158.  The  sum  of  all  the  edges  of  a  cube  is  36  in.     What  is 
the  volume  of  the  cube  ? 


CHAPTER   VTII 

BONDS  AND  STOCKS 
BONDS 

1.  When  a  government  or  a  private  corporation  is  in  need 
of  money  it  sometimes  borrows  it  and  issues  its  bonds  for  the 
amount.     A  Bond  is  a  promissory  note  issued  under  the  seal  of 
a  government  or  a  corporation. 

Mr.  A.  has  a  bond  issued  by  the  United  States  government 
which  promises  to  pay  $  1000  at  a  certain  time,  with  interest 
at  4%.  What  is  his  yearly  income  from  this  bond? 

2.  In  1898  when  the  government  wished  to  raise  money  to 
carry  on  the  war  with  Spain,  it  issued  bonds  payable  in  20  yr., 
with  interest  at  3%.    People  who  wished  to  lend  their  money  to 
the  government  made  application  for  bonds.     When  their  appli- 
cations were  granted,  they  sent  to  the  Treasury  Department  of 
the  United  States  the  payment  for  the  bonds,  and  received  in 
return  bonds  for  that  amount.     If  you  had  sent  $4000  asking 
for  $  500  bonds,  how  many  bonds  would  you  have  received  ? 

3.  In  September,  1898,  Mr.  Koss  bought  a  $  500  bond  of  the 
issue  of  1898  and  gave  it  to  his  daughter  Julia.     The  bond 
contained  these  words : 

The  United  States  of  America  are  indebted  unto  the  bearer  in  the  sum 
of  Five  Hundred  Dollars.  This  bond  is  issued  under  authority  of  an  Act 
of  Congress  entitled  "An  Act  to  provide  ways  and  means  to  meet  war 
expenditure,"  and  is  redeemable  at  the  pleasure  of  the  United  States 
after  the  first  day  of  August,  1908,  and  payable  August  1,  1918,  in  coin 
with  interest  at  the  rate  of  three  per  centum  per  annum  payable  quarterly 
in  coin  on  the  first  day  of  November,  February,  May,  and  August  in  each 
year.  The  principal  and  interest  are  exempt  from  all  taxes  or  duties  of 
the  United  States  as  well  as  from  taxation  in  any  form  by  or  under  State, 
municipal,  or  local  authority. 

When  did  the  first  payment  of  interest  become  due  ?  How 
much  was  paid  ? 

280 


BONDS  281 

4.  If  Miss  Ross  keeps  her  bond  until  it  becomes  payable, 
how  much  interest  will  she  receive  ?     How  many  interest  pay- 
ments ? 

5.  Mr.  Eoss  has  85  $100  bonds,  375  $  200  bonds,  and  875 
$  500  bonds,  all  of  the  issue  of  1898.     What  is  his  income  each 
quarter  from  these  bonds  ?     What  is  the  yearly  income  ? 

6.  Many  persons  wish  to  invest  their  money  in  United 
States  bonds.     Can  you  see  why? 

7.  As  there  was  a  great  demand  for  the  1898  bonds,  and  the 
issue  was  limited,  many  people  who  applied  for  bonds  could  not 
get  all  they  wanted  from  the  government.      They  therefore 
tried  to  buy  them  from  the  holders,  and  the  bonds  soon  rose  in 
price.     They  were  then  said  to  be  above  par,  or  at  a  premium. 
When  bonds  are  at  a  premium  of  4%,  a  $  100  bond  costs  $  104. 
When  bonds  are  at  a  premium  of  4%,  how  much  will  30  $100 
bonds  cost  ? 

8.  At  104,  or  at  a  premium  of  4%,  how  many  $100  bonds 
can  be  bought  for  $  936  ?     For  $  2600  ? 

9.  How  much  will  75  $  100  bonds  cost  at  105  ? 

10.  Find  the  cost  of  30  $  500  bonds  at  2%  premium. 

11.  When  bonds  are  offered  at  a  lower  price  than  their  face 
value  they  are  said  to  be  below  par,  or  at  a  discount.      When 
bonds  are  at  a  discount  of  2%,  one  $100  bond  can  be  bought 
for  $98.     At  2%  discount,  how  much  would  50  $100  bonds 
cost? 

12.  At  98  how  much  must  be  paid  for  20  $100  bonds  ? 

13.  At  98  how  many  $100  bonds  can  be  bought  for  $1274? 
For  $1666? 

14.  September  1,  1898,  Mr.  Kane  bought  at  par  a  $5000 
bond  of  the  issue  of  1898.     He  held  it  until  March  1,  1899,  and 
sold  it  at  a  premium  of  7%.     How  much  did  he  gain  ? 

Observe  that  Mr.  Kane  would  receive  the  interest  that  fell  due  while 
he  held  the  bond. 


282  BONDS  AND   STOCKS 

In  the  following  problems  the  value  of  a  bond  is  $  100,  unless  otherwise 
stated. 

15.  What  is  the  value  of  183  bonds  of  a  city  corporation  at 
92%?     If  the  purchaser  holds  them  long  enough  to  realize  two 
4%  interest  payments  on  them,  and  then  sells  them  at  90%, 
how  much  does  he  gain  by  the  transaction? 

16.  In  order  to  build  a  court  house  the  county  of  Accalama 
issued  2500  bonds  bearing  5%  interest.     One  half  of  them  were 
sold  at  par,  and  the  rest  at  102.      How  much  money  did  the 
county  receive  for  its  bonds  ? 

17.  Mr.  Harvey  bought  $15,000  worth  of  the   Accalama 
bonds  at  par,  and  held  them  seven  years,  receiving  his  interest 
annually.     At  the  end  of  the  eighth  year  the  county  refused  to 
pay  the  interest  on  them.     The  bonds  fell  to  70%,  at  which 
price  Mr.  Harvey  sold  them  to  Mr.  Norton.      Did  Mr.  Harvey 
gain  or  lose,  and  how  much? 

18.  The  next  month  after  Mr.  Norton  bought  the  bonds,  the 
county  of  Accalama  redeemed  them  at  95%.     How  much  did 
Mr.  Norton  gain  by  the  transaction? 

19.  In  order  to  raise  the  money  to  redeem  the  5%  bonds,  the 
county  of  Accalama  issued  the  same  amount  of  bonds  bearing 
3%  interest.     The  new  bonds  sold  at  93.      Mr.  Norton  bought 
150  bonds  at  this  price,  held  them  8  yr.,  and  sold  them  at  97%. 
How  much  did  he  gain  on  this  investment  ? 

20.  How  much  interest  did  the  county  save  each  year  by 
substituting  3%  bonds  for  5%  bonds? 

21.  Persons  who  buy  and  sell  bonds  for  others  are  called 
brokers.     They  are  paid  a  certain  per  cent  on  the  par  value  of 
the  bonds  bought  and  sold.    This  percentage  is  called  Brokerage. 

If  you  were  to  pay  a  broker  -|-%  for  buying  40  $100  bonds 
for  you,  how  much  brokerage  would  you  pay  ?  If  the  bonds 
were  at  par,  how  much  would  they  cost  you,  including  broker- 
age ?  If  the  bonds  were  below  par  would  the  brokerage  be  less  ? 


BONDS  283 

22.  Imagine  yourself  a  broker  receiving  %<f0  for  buying  or 
selling  bonds  for  others.     If  you  were  to  sell  70  $  100  bonds 
and  to  buy  90  $  500  bonds,  how  much  brokerage  should  you 
receive  ? 

23.  How  much  brokerage  should  you  receive  if  you  sold  30 
$  100  bonds,  40  $  1000  bonds,  and  a  $  5000  bond  ? 

24.  A  broker  sold  20  United  States  $  100  bonds,  300  Cass 
County  bonds,  par  value  $  50,  and  200  City  Improvement  bonds, 
par  value  $25  each.     His  brokerage  was  \%-     To  how  much 
did  it  amount  ? 

25.  A  broker  bought  for  a  client  40  railroad  bonds  at  87, 
and  40  United  States  bonds  at  109.     His  brokerage  was  £%. 
From  which  transaction  did  he  receive  the  more  brokerage, 
and  how  much  more  ? 

Observe  that  brokerage,  premium,  discount,  and  interest  are  all  com- 
puted on  the  par  value. 

26.  A  broker  bought  for  Mr.  X.  20  bonds  at  103,  charging 
\°Jo  brokerage.     How  much  did  the  bonds  cost  Mr.  X.  ? 

Since  the  market  value  of  each  bond  was  $  103,  and  the  brokerage  on 
each  bond  was  \  of  a  dollar,  the  purchasing  price  of  each  bond  was  $  103£. 
20  bonds  would  cost  20  times  $  103£,  or  $  2062.50. 

27.  How  much  must  be  paid  for  80  railroad  bonds  quoted  at 
77,  brokerage  J%  ? 

28.  Through  his  broker  Mr.  S.  invested  $  2619  in  bonds  at 
109,  paying  J%  brokerage.     How  many  of  these  bonds  did  he 
buy? 

29.  Buying   United   States   4's   at   111-J-,   and   paying   |% 
brokerage,  Mr.  S.  invested  $  100,046.25.      How  many  bonds 
did  he  receive  ?     What  was  his  annual  income  from  them  ? 

The  expression  "  United  States  4's  "  means  United  States  bonds  paying 
4%  interest. 


284  BONDS  AND  STOCKS 

30.  If  brokerage  is  £%,  how  much  money  would  be  needed 
to  make  the  following  investments  ? 

a  80  United  States  bonds  at  105. 
b   70  A.  and  X.  Eailroad  bonds  at  72. 
c  19  Memphis  bonds  at  89. 

31.  Mr.  X.  owned  20  bonds.    When  they  were  quoted  at  103, 
a  broker  sold  them  for  him,  charging  him  1%.     How  much 
did  he  receive  for  the  bonds  ? 

In  this  case  shall  the  brokerage  be  added  to  the  market  price  of  the 
bond,  or  subtracted  from  it  ?     Why  ? 

32.  Mr.  B.  had  90  bonds  issued  by  the  M.  and  Q.  E.  R.  Co. 
When  they  were  quoted  at  79,  his  broker  sold  them  for  him. 
How  much  did  he  receive,  brokerage  being  %%  ? 

33.  Mr.  N.  ordered  the  purchase  of  90  shares  at  87-J-.    When 
they  had  fallen  to  86J,  he  ordered  their  sale.     Brokerage  being 
\°/o  for  buying  and  \°f0  for  selling,  how  much  did  he  lose  ? 

34.  Mr.  James  obtains  30  bonds  quoted  at  85,  paying  a 
broker  -J-%  for  buying  them.     The  same  broker  sells  them  for 
Mr.  James  at  86 \.     Brokerage   |%.     How   much   does   Mr. 
James  gain,  and  how  much  does  the  broker  receive   for  his 
work  ? 

35.  Mr.  A.  bought  20  United  States  4's  at  108.     How  much 
did  they  cost  him  ?     How  much  interest  did  he  receive  from 
them  each  year  ? 

36.  Mr.  A.  invested  $7800  in  bonds  which  were  selling  at 
104.     How  many  bonds  did  he  buy  ?     If  they  paid  3%,  what 
yearly  income  would  he  receive  from  them  ? 

37.  When  United  States  4's  were  selling  at  111|  Mr.  A. 
invested  $  10,704  in  them.     How  many  bonds  did  he  buy,  and 
what  income  did  he  receive  from  them  ? 

38.  Find  what  yearly  income  can  be  derived  from  the  fol- 
lowing investments : 


STOCKS  285 

a  $8240  invested  in  5%  bonds  at  103. 

b  $  6755  invested  in  3%  bonds  at  96f 

c  $  1584  invested  in  5%  bonds  at  99. 

d  $20,400  invested  in  4%  bonds  at  102. 

STOCKS 

39.  Across  a  certain  river  in  Ohio  there  was  formerly  a  toll 
bridge  on  which  the  fare  for  foot  passengers  was  15^.     John 
Smart,  a  schoolboy,  thought  it  would  be  a  profitable  scheme  to 
buy  a  $10  skiff,  and  to  carry  passengers   across   the   river 
during  his  vacation,  for  10  $  each.     As  he  had  not  enough 
money  to  pay  for  the  skiff,  he  formed  a  plan  similar  to  a  busi- 
ness enterprise  called  a  stock  company,  in  which  his  father 
was  interested.     John  induced  several  of  his  friends  to  join 
him  in  buying  the  skiff.     They  agreed  that  John  should  row 
the  passengers  across  the  river  every  day  in  the  week,  except 
Sunday,  and  should  retain  in  return  for  his  labor  50  ^  a  day 
from  the  gross  receipts.     The  rest  of  the  money  received  was 
to  be  divided  among  the  owners  of  the  skiff  in  proportion  to 
the  amount  each  had  invested.     The  first  week  55  passengers 
were  carried  across.     After  deducting  John's  salary,  how  much 
remained  to  be  divided  among  the  owners  of  the  boat  ?     What 
per  cent  was  that  of  the  whole  capital  ? 

40.  At  that  rate  how  much  was  received  by  Albert  Blake, 
who  had  put  in  $  5  ?     By  Edgar  Howe,  whose  share  of  the 
capital  was  f  3  ?     By  Fred  Lee,  whose  investment  was  $  1  ? 

41.  John  had  put  in  $1.     How  much  did  he  receive  from 
his  investment  and  his  salary  ? 

42.  Find  the  per  cent  of  gain  on  capital,  and  the  amount 
received  by  each  boy  at  the  end  of  the  week  in  which  32  pas- 
sengers were  carried  across.     Of  the  week  in  which  69  passen- 
gers were  carried  across. 

43.  Through  John's  carelessness  the  boat  was  overturned 
one  day,  and  the  skiff  route  became  unpopular.     The  receipts 


286  BONDS   AND   STOCKS 

for  the  week  in  which  the  accident  occurred  were  only  $2. 
How  much  did  each  boy  have  to  contribute  to  pay  John's 
salary  of  50  $  a  day  ? 

44.  The  next  week  they  sold  the  skiff  for  60%  of  what  it 
cost.    What  per  cent  of  his  investment  should  each  boy  receive 
from  the  sale  ?     How  much  money  ? 

SUGGESTION  TO  TEACHER.  Before  taking  up  the  study  of  stocks  each 
pupil  should  comprehend  fully  the  principles  involved  in  the  problems 
about  John  and  the  skiff.  Let  pupils  make  similar  problems  by  imagin- 
ing different  happenings  to  John  and  his  companions. 

45.  As  large  business  enterprises  require  more  capital  than 
is  usually  owned  by  one  man,  it  is  common  for  many  persons 
to  unite  and  form  what  is  called  a  Stock  Company.    The  money 
with   which   the   company   carries   on   business   is  called  its 
Capital.      Each  member   of    the   company   is  called  a  Stock- 
holder. 

That  part  of  the  earnings  of  a  company  which  is  divided 
among  the  stockholders  is  called  a  Dividend.  Dividends  are 
computed  at  a  certain  per  cent  on  the  par  value  of  the  capital. 

The  par  value  of  A's  stock  is  $  5000.    Find  his  dividend  at  6  %  - 

46.  A  company  whose  capital  stock  is  $  500,000  distributes 
$  20,000  in  dividends.     What  is  the  rate  of  dividend  ?     What 
is  the  rate  of  dividend  When  it  distributes  $  30,000  ? 

47.  Mr.  Smart  is  a  stockholder  in  a  stock  company  called 
the  Ohio  Transportation  Co.,  which  runs  a  line  of  steamers. 
The  capital  stock  is  $  100,000,  and  it   is  divided  into  1000 
shares  of  $  100  each.     Mr.  Smart  owns  10  shares ;  Mr.  Howe, 
20  shares;   Mr.  Blake,  500  shares;   Mr.  Lee,  70  shares;  and 
the  remaining  shares  are  owned  by  others.     In  the  first  year 
of  its  existence  the  earnings  of  the  company,  after  paying  all 
the  expenses,  were  $  7000.     What  per  cent  of  the  capital  were 
the  earnings?     How  much   should  Mr.  Howe  receive?     Mr. 
Blake  ?     Mr.  Lee  ? 


STOCKS  287 

48.  Mr.  Smart  is  the  superintendent  of  the  line  of  steamers, 
and  receives  a  salary  of  $  5000  a  year.     What  was  his  income 
in  the  first  year  from  his  stock  and  his  salary  ? 

49.  The  next  year  the  Ohio  Transportation  Co.  paid  a  divi- 
dend  of    9%.     How   much   was   received  by   the   four   men 
mentioned  in  Ex.  47  ? 

50.  How  much  would  be  received  by  each  of  those  gentle- 
men if  the  dividend  were  12%  ?     5%  ?     7£%  ? 

51.  When  a  company  pays  a  large  dividend,  there  arises  a 
demand  for  its  stock,  and  its  shares  sell  above  par,  or  at  a 
premium. 

When  the  stock  of  the  Ohio  Transportation  Co.  reached 
104,  Mr.  Blake  sold  400  shares  of  it.  How  much  did  he  gain 
by  the  sale  ? 

52.  The  next  year  the  dividends  of  the  Ohio  Transporta- 
tion Co.  fell  to  2%.     How  much  dividend  was  received  by  Mr. 
.Howe  ?     Mr.  Blake  ?     Mr.  Lee  ? 

53.  How  much  did  Mr.  Smart  receive  from  the  company 
that  year  ? 

54.  The  next  year,  through  some  unfortunate  management, 
the  company  was  unable  to  pay  dividends.     Instead,  an  assess- 
ment of  5%  was  made  upon  each  share,  in  order  to  pay  the 
running  expenses  of  the  business.     How  much  was  paid  by 
Mr.  Howe  ?     Mr.  Blake  ?     Mr.  Lee  ? 

55.  Mr.  Smart's  salary  was  lowered  10%.     How  much  did 
he  receive  from  the  company  that  year  ? 

56.  The  price  of  the  shares  of  the  Ohio  Transportation  Co. 
had  fallen  to  67.     Mr.  Blake  bought  600  shares  at  that  price. 
Mr.  Smart,  Mr.  Howe,  and  Mr.  Lee  bought  50  shares  each  at 
the  same  price.     At  the   end  of  the  year  the  dividend  was 
10%.     What  was  the  amount  of  each  man's  dividend? 

Remember  that  the  dividend  is  always  reckoned  on  the  par  value — 
whatever  may  be  the  quotation  in  the  market. 


288  BONDS   AND   STOCKS 

57.  March  1,  1895,  Mr.  Eeed  bought  40  shares  of  A.  &  B. 
R.  R.  stock  at  87.     The  shares  paid  a  semiannual  dividend  of 
3£%.     He  sold  them  March  1,  1896,  at  88.     How  much  did  he 
receive  in  dividends  ?     What  was  his  profit  from  the  advance 
in  price  ? 

In  the  following  problems  the  par  value  of  a  share  of  stock  is  assumed 
to  be  f  100. 

58.  If  you  were  to  receive  100  shares  in  a  mining  company 
which   pays   an  average   semiannual   dividend  of  4%,  what 
would  be  your  yearly  income  from  those  shares  ? 

SUGGESTION  TO  TEACHER.     Get  blank  certificates  of  stock.    Let  pupils 
form  themselves  into  an  imaginary  stock  company. 

59.  How  much  must  be  paid  for  90  shares  of  A.  &  C.  R.  R. 
stock  at  97-J-  ?     If  these  shares  pay  a  semiannual  dividend  of 
4£%,  how  much  yearly  income  will  be  derived  from  them  ? 

60.  The   A.  &   B.   Belting   Co.,   whose   capital   stock  was 
$  500,000,  distributed  $  50,000  among  its  stockholders.     What 
was  the  rate  of  dividend  ?     How  much  was  received  by  Mr. 
Smith,  who  owned  37  shares  ? 

61.  At  the  time  the  above  dividend  was  declared,  banks 
were  paying  2%  interest  on  long  time  deposits.     Would  the 
shares  of  the  A.  &  B.  Belting  Co.  be  likely  to  be  at  par,  at  a 
premium,  or  at  a  discount  ? 

62.  How  much  must  be  paid  for  70  shares  of  the  A.  &  B. 
Belting  Co.,  at  105  ? 

63.  Mr.  A.  bought  900  shares  of  the  Unity  Coal  Mine  at  47. 
He  held  them  until  he  had  received  two  semiannual  dividends 
of  3%,  two  of  3£%,  and  three  of  4%.     He  then  sold  the  shares 
at  71.     How  much  did  he  gain,  including  the  dividends  ? 

64.  How  much  must  be  paid  for  315  shares  of  the  W.  U. 
Telegraph  stock  at  137£  ?     If  these  shares  yield  a  semiannual 
dividend  of  11         what  is  the  annual  income  from  them  ? 


STOCKS  289 

65  o  How  many  shares  of  N.  &  St.  L.  stock  quoted  at  93  can 
be  bought  for  $  7440  ?  If  these  shares  pay  an  annual  dividend 
of  5%,  what  is  the  annual  income  from  them  ? 

SUGGESTION.  Since  one  share  costs  $  93,  how  many  shares  will  $  7440 
buy  ?  On  what  is  the  dividend  reckoned  ?  What  is  the  par  value  of  the 
stock  ? 

66.  How  many  shares  at  95  can  be  bought  for  $  7600  ? 
What  is   the   annual   income   from  them   if    they   pay    6% 
dividend  ? 

67.  Find  annual  income  from  $  7254  invested  in  D.  &  H. 
R.  R.  stock  at  78,  the  semiannual  dividend  being  4%  ? 

68.  The  K.  &  X.  R.  R.  Co.  paid  a  dividend  of  8%  in  Janu- 
ary, and  another  of    7£%   in  July.      What  was  the  yearly 
income  of  a  stockholder  who  owned  750  shares  ? 

Find  annual  income  from  the  following  : 

69.  $  22,464  invested  in  stocks  at  108,  which  pay  12%. 

70.  $  6965  invested  in  stocks  at  99£,  which  pay  7%. 

71.  $  12,390  invested  in  stocks  at  88£,  which  pay 


72.  Mr.  A.  bought  a  share  of  stock  at  80,  which  paid  8%. 
What  per  cent  did  he  gain  on  his  investment  ? 

What  amount  of  dividend  did  Mr.  A  .  receive  ?     An  $  8  dividend  is 
what  per  cent  of  an  $  80  investment  ? 

Find  what   per  cent   is   gained   annually  on  the  following 
investments  : 

73.  Stocks  bought  at  50,  paying  2%. 

74.  Stocks  bought  at  70,  paying  3£%. 

75.  Bonds  bought  at  102£,  paying  3%. 

76.  Stocks  bought  at  87-j-,  paying  7%. 

77.  Stocks  bought  at  41f,  paying 

HORX.    GRAM.    SCH.    AR.  -  19 


290  BONDS  AND   STOCKS 

78.  In  the  case  of  a  person  who  does  not  have  to  pay  brok- 
erage, which  pays  the  better  per  cent,  and  how  much,  4% 
bonds  at  par  or  6%  shares  at  90  ? 

79.  Mr.  E.  bought  75  U.  S.  4's  at  110  and  75  shares  C.  & 
L.  E.  E.  stock  at  90  without  brokerage.     The  stocks  paid  5% 
dividend.     The  bonds  cost  how  much  more  than  the  stocks  ? 
How  much  more  income  did  he  receive  from  his  stocks  than 
from  his  bonds  ?     What  per  cent  did  he  make  on  each  invest- 
ment? 

Brokerage  being  |%  for  buying  and  the  same  for  selling, 
how  much  is  gained  or  lost  on : 

80.  90  shares  bought  at  70,  sold  at  83  ? 

81.  113  shares  bought  at  64,  sold  at  59£  ? 

82.  27  shares  bought  at  58,  sold  at  57}  ? 

83.  800  shares  bought  at  41  j,  sold  at  52  J  ? 

84.  70  shares  bought  at  112J,  sold  at  113  ? 

85.  900  shares  bought  at  102,  sold  at  lOlf  ? 

MISCELLANEOUS  EXERCISES 

1.  What  is  the  largest  prime  number  that  can  be  expressed 
by  three  figures  ? 

2.  Eesolve  54  into  prime  factors.     What  per  cent  of  them 
are  3's  ? 

3.  If  4-  of  the  price  of  a  ship  is  $  12,000,  how  much  is  the 
whole  ship  worth  ? 

4.  A  man  owning  f  of  an  estate  sells  f  of  his  share  for 
$  2400.     At  this  rate,  how  much  is  the  estate  worth  ? 

5.  Jane  is  8  yr.  old,  and  Lucy  13.     The  sum  of  Jane's  and 
Lucy's  ages  less  7  yr.,  is  the  age  of  Mary.     How  old  is  Mary  ? 


MISCELLANEOUS  EXERCISES  291 

6.  A  farmer  had  two  fields  of  wheat ;  the  first  yielded  840 
bu.,  which  was  -f%  of  the  amount  yielded  by  the  second.     How 
many  bushels  did  he  get  from  both  fields  ? 

7.  A  man  bought  a  firkin  of  butter  for  $  17,  a  crock  of  lard 
for  $  8,  and  a  barrel  of  flour  for  $  9.     To  pay  for  them  he 
needed  $  7.50  more  than  he  had.     How  much  money  had  he  ? 

8.  By  what  must  1.7  be  multiplied  to  make  5.95  ?     6.46  ? 

9.  John  rode  7-J  miles  on  his  bicycle  in  one  hour,  6T7^  in 
the  next  hour,  and  6|  in  the  next.     How  far  did  he  ride  in 
all  ?     How  much  farther  in  the  first  hour  than  in  the  second  ? 
Than  in  the  third  ? 

10.  Harry  walked  7.64  miles,  and  James  walked  twice  as  far. 
How  far  did  they  both  walk  ? 

11.  A  merchant  bought  a  barrel  of  sugar  for  $  28.50,  and  a 
barrel  of  flour  for  $  7.50.     He  sold  the  two  for  $  40.     What 
per  cent  did  he  gain  ? 

12.  How  many  millimeters  in  the  circumference  of  a  circle 
whose  diameter  is  7  centimeters  ? 

13.  The  circumference  of  a  wheel  is  2.6  m.     How  many 
times  will  it  revolve  in  rolling  33.8  m.  ? 

14.  How  many  square  decimeters  in  the  surface  of  a  stere  ? 

15.  How  many  steres  of  wood  in  a  pile  17  m.  long,  8  m. 
wide,  and  2  m.  high? 

16.  Find  the  cost  of  digging  a  cellar  7  m.  long,  5  m.  wide, 
and  2  m.  deep,  at  20  ^  a  stere. 

17.  Image  a  cubic  centimeter  of  water.     How  much  does 
it  weigh  ?      5£  liters   of   water   weigh   how  many  grams  ? 
Kilograms  ? 

18.  How  many  kilograms  will  7  liters  of  alcohol  weigh  if 
•alcohol  is  %  as  heavy  as  water  ? 

19.  If  ice  weighs  94%  as  much  as  water,  how  many  kilo- 
grams of  ice  are  there  in  a  block  of  ice  9  dm.  long,  5  dm.  wide, 
and  4  dm.  high  ? 


292  BONDS  AND   STOCKS 

20.  How  many  kilograms  do  2  liters  of  mercury  weigh, 
mercury  being  13.5  times  as  heavy  as  water? 

21.  The  product  of  two  numbers  is  f.     One  of  the  numbers 
is  2J.     What  is  the  other  number  ? 

22.  A  landowner  divided  1\  A.  of  land  into  city  lots  55  ft. 
in  front  and  132  ft.  deep,  first  taking  out  for  streets  and  alleys 
108,900  sq.  ft.     How  many  lots  were  there  ?     He  sold  them  at 
an  average  of  $  40  a  front  foot.     The  land  had  cost  him  $  100 
an  acre  20  yr.  before.     He  had  paid  an  average  of  $  300  a  year 
in  taxes  upon  it,  and  the  expense  of  platting  and  selling  it  was 
$  315.     How  much  did  he  gain  by  holding  the  land  ? 

23.  How  many  bricks  8  in.  by  4  in.  will  be  required  to  pave 
a  yard  168  ft.  long  and  60  ft.  wide  ? 

24.  If  Mr.  A.  were  to  lose  33^%  of  his  money,  he  would  have 
$  2000  left.     How  much  money  has  he  ? 

25.  What  is  the  value  of  a  pile  of  wood  30  ft.  long,  8  ft. 
wide,  and  4  ft.  high  at  f  3.75  a  cord? 

26.  \  of  8  is  what  per  cent  of  £  of  20  ? 

27.  Mrs.  A.  has  at  interest  $800  at  6%  and  $1000  at  5%. 
What  is  her  yearly  income  from  both  investments  ? 

28.  If  a  piano  which  cost  $  260  is  sold  at  $  325,  what  per 
cent  is  gained? 

29.  What  is  the  interest  of  $  4270  from  May  1,  1895,  to 
Aug.  1,  1901,  at  5%  ? 

30.  Mr.  A.  borrowed  $  7000  at  5%.     At  the  end  of  each  of 
the  first  two  years  he  paid  $  1000.     At  the  end  of  the  third 
year  he  paid  all  that  was  due.     How  much  did  he  pay  ? 

31.  A  note  for  $782.50  payable  in  60  da.  with  grace  was 
discounted  at  6%.     What  were  the  proceeds? 

32.  Mr.  0.  failed  in  business,  owing  $60,000  and  having* 
$30,000  with  which  to  pay.     What  per  cent  of  the  amount 
could  he  pay  ?     How  much  would  a  creditor  receive  to  whom 
he  owed  $  1800  ?     A  creditor  to  whom  he  owed  $  2456.65  ? 


MISCELLANEOUS  EXERCISES  293 


33.  If  by  selling  fruit  at  9^  a  pound  a  grocer  gains 
how  much  will  he  gain  by  selling  it  at  11  $  a  pound  ? 

34.  What  per  cent  would  a  jeweler  gain  by  selling  a  watch 
at  $80,  if  by  selling  it  at  $  75  he  gains  50%  ? 

35.  A  merchant  bought  goods  at  $  1.6.0,  marked  them  to  sell 
at  an  advance  of  37-J%,  and  sold  them  at  a  reduction  of  25%  on 
the  marked  price.     At  what  price  were  they  sold  and  what  per 
cent  was  gained  on  them  ? 

36.  Goods  costing  $  864  were  marked  at  an  advance  of  50% 
and  sold  at  a  discount  of  16  f  %  from  the  marked  price.     How 
much  was  gained  on  them  ?     What  per  cent  ? 

37.  Find   the   net   cost   of  a  bill   of   goods   amounting   to 
$375.50  with  discounts  of  60%,  40%,  and  5%. 

38.  Make  a  problem  which  involves  trade  discount. 

39.  An  agent  sold  250  bbl.  of  flour  at  $  3.80  per  barrel,  com- 
mission 3%.      What  was  his  commission,  and  how  much  was 
sent  to  the  owner  of  the  flour  ? 

40.  A  steamer  valued  at  $  750,000  was  insured  for  f  of  its 
value  at  1£%,  in  two  companies,  one  company  taking  \  of  the 
risk  and  the  other  the  remainder.     What  was  the  amount  of 
premium  for  each  company  ? 

41.  Mr.  A.  has  real  estate  assessed  at  $  20,000  and  personal 
property  to  the   amount  of  $  8224.     He  pays  a  poll  tax  of 
$1.50.     What  is  the  amount  of  his  taxes  when  the  rate  of 
taxation  is  37^  mills  on  a  dollar  ? 

42.  What  is  the  duty  on  20  casks  of  wine,  each  cask  con- 
taining 56  gal.,  invoiced  at  $  2.35  per  gallon,  if  12^%  is  allowed 
for  leakage  and  if  there  is  an  ad  valorem  duty  of  45%  ? 

43.  Find  the  cost  of  80  shares  1ST.  Y.  C.  E.  E.  stock  at  112, 
brokerage,  1%. 

44.  How  much  must  one  pay  for  65  U.  S.  4's  at  108,  broker- 
age, 


294  BONDS  AND  STOCKS 

45.  How  much  would  one  receive  from  selling  30  shares  of 
mining  stock  at  91,  brokerage,  -J%  ? 

46.  How  much  does  the  owner  receive  from  the  sale  of  95 
shares  of  stock  sold  at  103,  brokerage,  %%? 

Brokerage  being  -J%,  for  buying  and  for  selling,  how  much  is 
gained  or  lost  by 

47.  Buying  72  shares  at  89  and  selling  them  at  90? 

48.  Buying  40  shares  at  71£  and  selling  them  at  70? 

49.  Buying  35  shares  at  59  and  selling  them  at  58 J? 

50.  At  57|-  how  many  shares  of  stock  can  be  bought  for 
$7156.25,  brokerage,  1%  ?     If  they  pay  a  semiannual  divi- 
dend of  4%,  what  income  is  derived  from  them  ? 

51.  What  income  is  derived  from  $  15,300  invested  in  stocks 
at  95£,  brokerage,  -J%,  if  the  stocks  pay  a  semiannual  dividend 
of  3£%  ? 

52.  A  blacksmith's  price  for  shoeing  a  horse  was  50^  a  shoe, 
but  he  allowed  a  discount  of  10  °/o  to  any  person  bringing  him 
10  or  more  horses  at  the  same  time.     Mr.  Boyd's  horses  were 
shod  there  one  day  at  a  cost  of  $  13.50.    How  many  horses  were 
there? 

53.  A  room  18  ft.  by  15  ft.  was  covered  with  matting  1  yd. 
wide,  at  a  cost  of  $10.50.     To  lay  the  matting  cost  5^  a  yard. 
What  was  the  price  per  yard  of  the  matting? 

54.  The  Troy  Edge  Tool  Works  sold  12  doz.  sledge  ham- 

mers, weighing  5  Ib.  each,  at  10^  per  lb.,  and 
9   doz.   hammers   at  50^  apiece.      Discount, 
10.     Make  out  the  bill. 


55.  A  triangle  whose  sides  are  all  unequal 
is  called  a  Scalene  Triangle. 

Draw  a  right  triangle  whose  base  is  3  in. 
and  altitude,  4  in.  Is  it  scalene  ?  Give  rea- 
sons for  your  answer. 


MISCELLANEOUS  EXERCISES  295 

56.  Construct  an  equilateral   triangle.     An  isosceles   tri- 
angle.    A  scalene  triangle. 

57.  Find  the  perimeter  of  a  scalene  triangle,  of  which  one 
side  is  4£  in.,  another  side  is  6|  in.,  and  the  third  side  is  8f  in. 

58.  Find  perimeter  of  the  scalene  triangle  of  which  the  side 
AB  is  5  in.,  BG  is  2£  in.  longer  than  AB,  and  CA  is  3 3  in. 
longer  than  BC.     Represent. 

59.  How  long  is  the  perimeter  of  the  triangle  ABC,  when 
AB  is  12  in.,  BC  is  33£%  longer  than  AB,  and  CA  is  25% 
longer  than  BC  ? 

60.  Tell  how  a  line  is  bisected.     In  the  same  way  bisect  an 
arc. 

61.  If  the  base  of  an  isosceles  triangle  is  16  in.,  and  if  each 
of  the  other  sides  equals  87£%  of  the  base,  how  long  is  the 
perimeter  ? 

62.  At  $3£  per  day  for  board,  how  many  days  can  a  man 
board  at  the  seashore  for  $28? 

63.  A  man  had  three  lots,  each  containing  6£  A.,  which  he 
redivided  into  building  lots  of  f  of  an  acre  each.      How  many 
building  lots  did  he  have  ? 

64.  Mrs.  A.  wishes  to  cover  the  floor  of  a  room  16  ft.  long 
and  12  ft.  wide  with  ingrain  carpet  1  yd.  wide.     The  carpet 
will  cut  to  the  best  advantage  if  the  strips  are  laid  lengthwise. 
One  pattern  requires  that  the  shortest  possible  strips  shall  be 
16  ft.  8  in.  long.     This  carpet  costs  73^  per  yard.     Another 
pattern  requires  that  the  strips  shall  be  only  16  ft.  2  in.  long, 
but  it  costs  75^  a  yard.     What  is  the  difference  in  the  cost  of 
the  carpets  ? 

65.  How  much  more  will  it  cost  to  cover  a  floor  21  ft.  long 
and  18  ft.  wide  with  Brussels  carpet  27  in.  wide  at  $  1.25  per  yard 
than  with  a  yard  wide  ingrain  at  85  ^,  if  the  Brussels  requires 
only  1  in.  to  be  turned  in  at  the  end  of  the  strips  and  the 
ingrain  requires  6  in.  ?     The  strips  run  lengthwise. 


CHAPTER  IX 
LITERAL   QUANTITIES 

1  .    If  x  =  10,  how  many  days  in  x  weeks  ?      How  many 
minutes  in  x  hours  ?     How  many  cents  iff  x  dollars  ? 

2.  If  x  =  48,  how  many  yards  in  x  ft.  ?     Years  in  x  mo.  ? 
Ounces  in  x  Ib.  ?     Pecks  in  x  qt.  ?     Gallons  in  a?  qt.  ? 

3.  If  6  represents  3,  how  much  is  3  times  b  or  3  b  ? 

4.  If  a  =  8,  how  much  is  3a  ?     Jaor-?     2J  times  a? 

5.  If  x  =  18,  how  much  is  .5  x  ?     .7  x?     fa;?     33|%  of  a? 

6.  If  a  =  12,  how  much  is  25%  of  a  ?     16f  %  of  a  ?     12J% 
of  a? 

7.  When  a;  =  20,  how  much  is  —  ?     —  ?     -?     —  ? 

10       30       5        5 

8.  x  dollars  +  y  dollars  =  how  many  dollars,  when  x  =  8  and 


9.    If   a  =  10  and  6=3,  how  much  is  a  +  b?     3  a  +  ob? 
6b-a?     Sb-2a?     a  +  b?     b  +  a? 


10.  Give  some  values  to  x  and  y  that  will  make  the  follow- 
ing equations  true  :  #  +  ?/  =  19.     x-\-y  =  15.     x  —  y  =  \. 

11.  If  an  orange  costs  3^,  how  much  will  x  oranges  cost, 
when  x  =  10  ?     When  x  =  4  ? 

12.  How  many  oranges  in  a;  doz.  when  $  =  7  ? 

13.  If  ic  =  25,  how  many  weeks  in  the  number  of  days  repre- 
sented by  a;  +  3  ?     a  +  10?     z-4?     2x-l? 

296 


LITERAL  QUANTITIES  297 

14.  If  we  let  x  stand  for  35,  how  many  gallons  are  there  in 
the  number  of  quarts  that  are  represented  by  x  -f  1  ? 

15.  When  x  represents  14,  how  many  feet  in  the  number  of 
yards  represented  byoj-f-1?     a?  -f  3  ? 

The  expression  ab  means  a  times  6,  just  as  4  b  means  4  times  b. 

16.  If  a  =  7  and  b  =  5,  how  much  is  ab?     Which  is  the 
greater,  7  times  5  or  5  times  7  ?     a  times  b  or  b  times  a  ? 

17.  If  a  =  11  and  6  =  2,  how  much  is  3  ab  ?    3ba  ? 

18.  How  much  will  x  apples  cost  at  y$  a  piece,  if  x  =  4  and 
y  =  2  ?     If  z  =  8  and  #  =  3  ? 

19.  If  we  represent  the  cost  of  one  apple  by  y  and  a  number 
of  apples  by  aj,  how  shall  we  represent  the  cost  of  them  all  ? 

20.  Give  some  values  to  x  and  y  that  will  make  the  follow- 
ing equations  true.     #i/  =  48.     xy  =  35.     xy  =  70.-    xy  =  98. 

21.  When  x  =  50  and  y  =  2,  how  many  oranges  can  be  bought 
for  xtf,  if  one  orange  costs  ytf?     How  many  when  #  =  30  and 


22.  If  we  represent  the  cost  of  one  orange  by  y  and  the  cost 
of  a  number  of  oranges  by  x,  how  shall  we  represent  the  num- 
ber of  oranges  ? 

23.  If  we  represent  the  cost  of  one  orange  by  x  and  the  cost 
of  a  number  of  oranges  by  y,  how  shall  we  represent  the  num- 
ber of  oranges  ? 

24.  Give  some  values  to  x  and  y  that  will  make  the  follow- 
ing equations  true  : 

2.7.         2  =  4.         *  =  6.         Z  =  9.         5  =  3. 

y  »  y  a;  2/ 

25.  3(5  +  4)  =  ? 

This  expression  means  "3  times  the  sum  of  5  and  4."     Quantities 
inclosed  in  a  parenthesis  are  to  be  considered  as  one  quantity. 


298 


LITERAL  QUANTITIES 


26.    2(7  +  2)  =  ?     2(7-2)  =  ?     8(6  +  4)  =  ?     (7  +  3)  =  ? 

o 


27.  If  a  =7  and  6  =  2,  how  much  is  2  (a  +  6)?    (a  +  6)2? 

(a-6)2?    2(o-6)?    (a  +  5)?    (a~6)? 
3  5 

28.  When  a  =  4  and  6  =  3,  how  much  is  5  (a  +  6)  ? 
2(a-6)?     a(a  +  6)?     6(a  +  6)?     (a  +  6)2?     (a-6)2? 

29.  When  £  =  10  and  u  =  3,  how  much  is 

3*  +  5w?        w(2*  +  u)?        t2  +  2tu  +  u2?        (t  +  u)2? 

30.  If  a;  is  3,  then  9  x  —  5  x  =  what  number  ? 

Finish  the  following  equations,  supposing  x  to  equal  7  : 


•          35. 

:          36.    44  —  5x  = 

37.  In  the  broken  line  ABODE 
the  part  BC  is  twice  as  long  as 
AB,  CD  is  3  times  as  long  as  AB, 
and  DE  is  4  times  as  long  as 
AB.  How  long  is  the  entire  line 
if  x  represents  7  in.?  2  ft.? 
5  yd.? 

38.  Draw  a  broken  line  consisting  of  two  parts  in  which  one 
part  is  4  times  as  long  as  the  other.  Let  x  stand  for  the  length 
of  the  smaller  part.  What  will  represent  the  length  of  the 
other  part  ?  How  long  is  the  entire  line  if  x  =  2  in.  ?  8  in.  ? 

A 3# B  39.   How   long  is   the   perimeter   of 

V~  ~A^    the  rhomboid  AB  CD  if  x  =  5  in.  ?     If 


FIG.  1. 


FIG.  2. 


40.  Draw  a  rhomboid,  making  a  longer 


LITERAL   QUANTITIES 


299 


side  twice  as  long  as  either  of  its  adjacent  sides.  Mark  a 
short  side  x  and  the  other  sides  accordingly.  How  long  would 
B  the  perimeter  of  the  rhomboid  be  if 

x=Sin.?     1.1  in.?     2J  in.  ? 

41.  In  the  circle  whose  center  is  0, 
the  arcs  are  in  the  ratios  represented 
in  Fig.  3.  How  long  is  the  circumfer- 


ence  if  x  =  l  in.  ? 


ft.? 


42.  How   long   is   the   perimeter   of 
the   trapezoid   DEFG,   if   x  =  10   in.? 
3f  in.  ? 

43.  8  a—  6a  +  3a  =  how  many  a's  ? 
8a-6a  =  2a;  2  a  +  3  a  =  5  a. 


Express  in  one  term 
44.    96-76  +  46. 


45. 
46. 
47. 


c-8c. 


+  2x-5x-l3x. 


48. 
49. 
50. 
51. 


4Sx-3x-llx-20x. 


/D  52.  In  the  broken  line,  ABCD, 
the  part  BC  is  twice  as  long  as 
AB,  and  CD  is  3  times  as  long 
as  AB.  The  entire  line  is  how 
many  times  as  long  as  AB  ? 

If  we  know  the  length  of  the 
entire  line,  we  may  find  by  equa- 
tions the  length  of  each  part.  If  the  entire  line  is  12  in. 
long,  we  have, 

x  +  2  x  +  3  x  =  12  in. 
then  6  x  =  12  in. 

and  x  =  2  in.,  length  of  AB. 

and  2  x  =  4  in.,  length  of  BC. 

and  3  x  =  6  in.,  length  of  CD. 


300  LITERAL   QUANTITIES 

53.  Find  the  length  of  each  part  of  the  broken  line  repre- 
sented in  Fig.  5,  if  that  line  is  30  in.  long.  54  in. 
long. 

54.  In  the  isosceles  triangle,  ABC,  each  of  the 
equal  sides  is  twice  the  base.     The  perimeter  is 
45  in.     How  long  is  each  side  ? 

55.  Construct   an   isosceles   triangle,   in   which 
each  of  the  equal  sides  is  3  times  the  base.     Let 

FIG  6  x  =  the  base,  and  find  how  long  each  side  would  be 
if  the  perimeter  were  56  in.  105  in.  147  in. 

56.  In   the   trapezoid,   ABCD,   the   non-parallel   sides   are 

equal,  the  upper  base  is  3  times  as  long 

•£• ^ 5       as  either  of  its  adjacent  sides,  and   the 

2_ \  lower  base  is  4  times  as  long  as  either 

0  of  its  adjacent  sides.     How  long  is  each 
side,  if  the  perimeter  is  45  in.  ?     153  in.  ? 

57.  Turn  to  Fig.  4,  page  299,  and  find  the  length  of  each 
side  of  the  trapezoid,  if  its  perimeter  is  42  in.     9  ft.  4  in. 

58.  In  the  trapezium,  ABCD,  the  sides  have 
to  one  another   the   ratios  expressed  in  Fig.  8. 
How  long  would  each  side  be  if  the  perimeter 
were  80  in.  ?     75  in.  ? 

59.  Mr.  Morton  spent  some  money  on  Monday, 
3  times  as  much  on  Tuesday,  and  5  times  as  much 
on  Wednesday.      If  he  spent  $  36  in  all,  how 
much  did  he  spend  each  day  ? 

Let  x  =  the  number  of  dollars  spent  on  Monday. 

60.  John  has  4  times  as  many  marbles  as  James,  and  they 
both  have  75  marbles.     How  many  has  each  ? 

61.  The  sum  of  two  numbers  is  21,  and  one  of  them  is  6 
times  the  other.     What  are  the  numbers  ? 


LITERAL   QUANTITIES  301 

62.  Ella's  mother  is  3  times  as  old  as  Ella.     Her  father  is 
4  times  as  old  as  Ella.     The  sum  of  all  their  ages  is  96  years. 
How  old  is  each  ? 

63.  I  am  thiirking  of  two  numbers,  one  of  which  is  5  times 
the  other.     Their  sum  is  18.     Find  the  numbers. 

64.  CLASS  EXERCISE.          -  may  think  of  two  numbers,  one 
of  which  is  a  multiple  of  the  other.     He  may  give  the  sum  and 
the  ratio  of  these  numbers  and  the  class  may  find  them. 

65.  The  circumference   represented  is 
54  ft.     The  arc  AB  is  twice  the  arc  BC, 
and  the  arc   CA  is  three  times  the  arc 

\c  BC.     How  long  is  each  arc  ? 

66.  Turn  to  Fig.  3,  page  299.    Find  the 
length  of  each  arc   when  the  circumfer- 

FIG.  9.  ence  is  96  ft.     When  it  is  8  ft.  4  in. 

67.  Ida  set  out  a  number  of  geraniums,  twice  as  many  roses 
as  geraniums,  and  three  times  as  many  pansies  as  roses.  There 
were  27  plants  in  all.  How  many  were  there  of  each  ? 

Let  x  =  the  number  of  geraniums, 

then  2x=   "          "         "   roses, 

and  6x=  u         "        "  pansies. 


68.  There  are  three  numbers  whose  sum  is  80.     The  second 
is  3  times  the  first,  and  the  last  is  4  times  the  second.     Find 
them. 

69.  Ida,  Frank,  and  Henry  paid  $  10  to  have  a  tennis  court 
prepared.     Frank  gave  three  times  as  much  as  Ida,  and  Henry 
gave  twice  as  much  as  Frank.     How  much  did  each  give  ? 

70.  On  the  day  that  Euth  Owen  was  22  years  old  she  re- 
ceived a  bunch  of  roses  consisting  of  one  rose  for  each  year  of 
her  life.     There  were  twice  as  many  pink  roses  as  red  roses, 


302  LITERAL  QUANTITIES 

and  four  times  as  many  white  roses  as  pink  roses.    How  many 
were  there  of  each  ? 

71.  Make  similar  problems. 

72.  A  certain  number  plus  itself  equals  320.     What  is  the 

number  ? 

x  +  x  =  320. 

73.  Find  the  number  which  added  to  itself  equals  258.    237. 

74.  A  certain  number  plus  twice  itself  equals  396.    Find  the 
number. 

75.  Find  the  number  which  added  to  twice  itself  equals  297. 

76.  Find  the  number  which  added  to  four  times  itself  equals 
195.     275.     300.     177. 

77.  Separate  18  into  two  parts,  one  of  which  is  8  times  the 
other. 


78.  Separate  30  into  two  parts,  one  of  which  is  5  times  the 
other.     One  of  which  is  9  times  the  other.    One  of  which  is  14 
times  the  other. 

79.  Separate  24  into  two  parts,  one  of  which  is  twice  the 
other.     3  times  the  other.     5  times  the  other. 

80.  CLASS  EXERCISE.     -  may  name  a  number  which  he 
can  separate  into  two  parts  whose  ratio  is  a  whole  number.   He 
may  give  the  ratio  of  those  parts.    The  class  may  find  the  parts. 

81.  Two  brothers,  Messrs.  Arthur  and  Philip   Owen,  paid 
$  18,000  for  a  piece  of  land.     Mr.  Arthur  Owen  paid  5  times 
as  much  as  his  brother.     How  much  did  each  pay  ? 

82.  What  number  is  that  to  which,  if  4  times  itself,  and  6 
times  itself  be  added  the  sum  is  77  ? 

83.  A  farmer  sold  a  cow  and  a  pig  for  $  30,  receiving  9  times 
as  much  for  the  cow  as  for  the  pig.    What  was  the  price  of 
each? 


LITERAL   QUANTITIES  303 

84.  Separate  60  into  three  parts  such  that  the  second  is  4 
times  the  first,  and  the  third  is  5  times  the  first. 

85.  What  number  added  to  six  times  itself  equals  147  ? 

86.  Albert  has  3  times  as  many  marbles  as  James.     Roy 
has  as  many  marbles  as  both  the  other  boys  have.     They  all 
have  72  marbles.     How  many  has  each  ? 

87.  Thomas  caught  3  fish.      The  largest  fish  weighed  as 
much  as  the  other  two.     One  of  those  weighed  twice  the  other. 
The  weight  of  all  was  6  Ib.     How  much  did  each  weigh  ? 

88.  One  hundred  can  be  separated  into  3  integers,  of  which 
the  second  is  4  times  the  first,  and  the  third  is  equal  to  the 
sum  of  the  first  and  second.     What  are  the  numbers  ? 

x  „+  4x  +  (x  4-  4z)  =  100. 

89.  In  the  same  way  separate  120.     150.     600. 

90.  AB  is  a  diameter.     The  arc  EC 
is  3  times  the  arc  AC.     The  arc  BA  is 
72  in.     How  long  is  the  arc  AC?    BC? 
The  circumference  ? 

91.  If  the  circumference  of  the  circle 
in  Pig.  10  were  112  in.,  how  long  would 

FiiTTo.  be  the  arc  AC?    AB? 

92.  Seven  times  a  certain  number,  minus  three  times  that 
number,  equals  24.     What  is  the  number  ? 

Let  x  =  the  number, 

then  Ix  —  3x  =  24. 

93.  Six  times  a  certain  number  —  4  times  that  number  =  10. 
What  is  the  number  ? 

94.  If  a  certain  number  is  multiplied  by  7,  and  if  the  same 
number  is  also  multiplied  by  5,  the  difference  between  those 
products  is  16.     Find  the  number. 

95.  The  difference  between  the  8th  multiple  and  the  5th 
multiple  of  a  certain  number  is  21.     Find  the  number. 


304 


LITERAL   QUANTITIES 


96.    CLASS  EXERCISE. 


may  think  of  a  number  and  of 


two  of  its  multiples.  He  may  tell  the  class  which  multiples 
they  are  and  the  difference  between  them.  The  class  may 
find  the  number. 

97.    John  picked  3  times  as  many  quarts  of  berries  as  his 
sister  picked.     He  picked  8  more  quarts  than  she  did.     How 
many  quarts  did  each  pick  ? 

98.  Mr.   Bond   drew  a  sum  of   money  from  the 
bank  on  Friday,  and  five  times  as  much  on  Satur- 
day.    He  drew  $  128  more  on  Saturday  than  on  the 

r  previous  day.     How  much  did  he  draw  each  day  ? 

99.  The  sides  of  the  triangle  ABC  are  in  the  ratios 
expressed  in  Fig.  11.     The  sum  of  AB  and  BC  is 

FIG.  11.      77  in.  more  than  AC.     How  long  is  each  side  ? 

s  100.   In  the  rhomboid  ABCD,  the  side 

AB  is  three  times  the  side  BC,  and  it  is 
10  in.  longer  than  BC.    How  long  is  the 
FIG.  12.  perimeter? 


101.  The  arc  BC  is  twice  as  long  as  the 
arc  AB,  and  it  is  8  in.  longer  than  AB. 
How  long  is  AB?  BC?  AOC  is  a  right 
angle  formed  by  radii.  How  long  is  the 
circumference  ?  The  diameter  ? 


102.  In  the  trapezium  ABCD,  AB  =  BC  and 
CD  =  AD.  AD  =  3  times  AB,  and  it  is  6  in. 
longer  than  AB.  Find  the  perimeter  of  the 
trapezium. 


FIG.  14. 


LITERAL   QUANTITIES  305 

103.  In  the  trapezoid  ABCD,  the  side 
DC  equals  4  times  AB,  and  is  9  in.  longer 
than  AB.     If   each   of  the   non-parallel 

D  c    sides  is  7  in.,  how  long  is  the  perimeter  ? 

104.  If  John's  shoes  cost  twice  as  much 

as  his  hat,  and  they  both  cost  $  3.60,  how  much  will  each 
cost? 

In  solving  this  problem  it  will  be  more  convenient  to  let  x  equal  the 
number  of  dollars  that  John's  hat  costs  than  to  let  x  equal  the  number 
of  dollars  that  his  shoes  cost.  Do  you  see  why  ? 

105.  What  number  subtracted  from  3  times  itself  gives  for  a 
remainder  14  ?     26  ?     32  ? 

106.  Mary  has  twice  as  many  books  as  Alice,  and  together 
they  have  36  books.     How  many  has  each  ? 

107.  John  and  William  together  have  eighty  marbles,  and 
John  has  7  times  as  many  marbles  as  William.     How  many 
has  each  ? 

108.  Albert  walks  3  times  as  far  east  from  a  certain  point  as 
John  walks  west  from  the  same  point.     They  are  then  80  ft. 
apart.     How  far  does  each  walk  ? 

109.  A  pole  12  ft.  long  is  sunk  in  the  water  so  that  the  part 
below  the  surface  is  3  times  as  long  as  the  part  above.     How 
much  is  below  the  surface  ? 

110.  A  tree  60  ft.  high  is  broken  so  that  the  part  which  has 
fallen  down  is  5  times  as  long  as  that  which  remains  standing. 
Find  height  of  the  stump. 

111.  Mr.  Colton  had  a  sum  of  money  at  interest  at  5%  and  a 
sum  twice  as  large  at  6%.     In  all  he  had  $600  at  interest. 
Find  how  much  he  had  at  each  rate  and  his  yearly  income  from 
both  principals. 

112.  3  x  =  8  +  7.     Find  the  value  of  x. 

HORN.    GRAM.    SCH.    AR. — 20 


306  LITERAL   QUANTITIES 

113.  If  we  have  the  equation  "4  a? +  7  a;  =  52 +  3,"  how  is 
the  equation  "  11  x  =  55  "  obtained  from  it  ? 

Uniting  quantities  of  the  same  kind  on  the  same  side  of  the 
equation  is  called  collecting  the  terms. 

Find  value  of  the  literal  quantity  in  each  of  the  following : 

114.  2x  +  5z-7 

115.  5 

116.  6y-4y  =  21-6-6. 

117.  9  x  H-  2  x  =  50  -4  +  20. 

118.  18  y  -  5  y  +  7  y  =  30  -  10  +  24  +  36. 

119.  The  terms  written  before  the  sign  of  equality  form  the 
first  or  left-hand  member  of  the  equation.     Those  written  after 
the  sign  of  equality  form  the  second  or  right-hand  member  of 
the  equation. 

How  many  terms  are  there  in  the  second  member  of  the 
equation  in  Ex.  118  ?  In  the  left-hand  member  ? 

120.  The  number  which  shows  how  many  times  the  literal 
quantity  is  taken  is  called  the  Coefficient  of  that  quantity.     In 
the  expression  11  x,  11  is  the  coefficient. 

Supply  missing  coefficients  in  the  following  equations, 
assuming  that  x  =  3.  ?o?  =  24.  ?o?  =  21:  a?+?cc  =  24. 

121.  Make  an  equation  containing  a  literal  quantity  whose 
coefficient  is  5. 

122.  If  a  =  b,  is  it  true  that  a  +  7  =  b  +  7?     Illustrate. 

An  equation  is  like  a  pair  of  scales  and  the  sign  of  equality  is  like 
the  beam  of  the  scales.  If  a  pound  is  added  to  one  side,  what  must  be 
added  to  the  other  side,  in  order  to  keep  it  balanced  ?  If  the  amount 
on  one  side  is  doubled,  what  must  be  done  to  keep  the  balance  ? 

123.  Add  7  to  both  members  of  the  equation  40  =  40.    Is  the 
resulting  equation  true  ? 

124.  If  7  is  added  to  one  member  of  an  equation  and  9  to 
the  other  member,  is  the  resulting  statement  true  ?     Illustrate. 


LITERAL   QUANTITIES  307 

125.  Write  an  equation.     Subtract  the  same  quantity  from 
both  members  of  it  and  show  whether  or  not  the  members  are 
still  equal. 

126.  Multiply  both  members  of  the  equation  8  =  8  by  the 
same  quantity.     Are  the  members  still  equal  ? 

127.  If  both  members  of  an  equation  are  divided  by  the 
same  quantity,  how  is  the  equation  affected  ?     Illustrate. 

128.  If  a  —  5,  is  it  true  that  a2  =  25?     What  has  been  done 
to  each  member  of  the  original  equation  ? 

129.  Illustrate  by  numbers  the  truth  of  the  statement,  "If 
the  same  operation  is  performed  upon  each  member  of  an  equa- 
tion the  members  are  still  equal." 

130.  Find  the  value  of  x  when  x  —  2  =  6. 

SOLUTION 

x  -  2  =  6. 

Adding  2  to  each  member    £-2  +  2=6  +  2. 
Hence,  x  =  6  +  2. 

or  x  =  8. 

What  was  the  purpose  of  adding  2  to  the  left-hand  member  ?  To  the 
right-hand  member  ? 

Compare  the  first  and  third  equations  in  the  above  solution.  It  will  be 
seen  that  in  the  first  equation  "2"  is  written  in  the  left-hand  member 
and  has  the  minus  sign,  while  in  the  third  equation  "2  "  is  in  the  right- 
hand  member  and  has  the  plus  sign. 

131.  Find  the  value  of  x  when  x  +-  3  =  12. 

SOLUTION 
x  +  3  =  12. 

Subtracting  3  from  each  member  x  +  3  —  3  =  12  —  3. 
Hence,  x  =  12  —  3. 

or  x  =  9. 

Why  should  we  here  subtract,  instead  of  adding  3  to  each  member  ? 
Compare  the  first  and  the  third  equations.      What  change  has  been 
made  in  the  first  to  produce  the  third  ? 


308  LITERAL  QUANTITIES 

132.  Changing  a  quantity  from  one  side  of  an  equation  to 
the  other  is  called  transposing  the  quantity.     Illustrate. 

133.  Study  the  solutions  of  Exs.  130  and  131  until  you  see 
the  truth  of  the  following  principle  : 

A  quantity  may  be  transposed  from  one  side  of  an  equation  to 
the  other  if  the  sign  prefixed  to  the  quantity  is  changed  from  plus 
to  minus  or  from  minus  to  plus. 

When  no  sign  is  prefixed,  the  plus  sign  is  understood. 

134.  In  the  equation  2  a?  +  7  =  28  —  x,  if  x  is  transposed 
what  sign  will  it  have  ?     What  sign  will  7  have  if  transposed  ? 

What  is  the  purpose  of  transposing  quantities  ? 

It  may  help  you  to  remember  to  transpose  correctly  if  you  repeat 
"  When  I  change  the  side  I  change  the  sign." 

Find  the  values  of  the  literal  quantities  : 

135.  x  —  7  =  23.  139.      3  x  +  7  =  25. 

136.  x  -  8  =  21.  140.      7  x  -  3  =  67. 

137.  5  x  —  5  =  50.  141.      4  y  +  3  =  51. 

138.  5^-4  =  35.  142.    12x  +  7  =  67. 

143.  CLASS  EXERCISE.     -  may  think  of   some   number, 
call  it  x,  and  make  an  equation  like  the  above  for  the  class  to 
solve. 

144.  Finding  the  value  of  the  literal  quantity  in  an  equa- 
tion is  called  solving  the  equation. 

Solve 


Transposing,  we  have  12  x  —  8  se  =  31  —  7. 

Collecting  the  terms,  we  have  4  x  =  24. 

Dividing,  we  have  x  =  6. 

Solve  the  equations: 

145.  5a  +  9  =  3x  +  17.  148. 

146.  10x  +  8  =  32-2x.  149.    9£-7a+4  =  .x-f  8. 

147.  7/-9  =  5    +  7,  150. 


LITERAL  QUANTITIES 


151.  15y-21  =  14y-37.  153. 

152.  15a+9-12x=25-2a.  154.    23^-24=48-  11  a?. 

155.  Complete  the  equation  4  x  +  ?  =  35,  when  as  =  8. 

156.  Complete  the  equation  7  ?/  +  5  =  30  +  ?,  when  2/  =  9. 

157.  If  a?  =  10,  is  the  equation  5  x  +  7  •  =  54  true  ? 

158.  Substituting  the  value  of  the  unknown  quantity  in  an 
equation  and  thus  proving  the  truth  of  the  equation  is  called 
verifying  the  equation. 

Solve  and  verify  the  equation  x  +  9  =  15. 

Solve  and  verify  : 

159.  a  +  7  =  21.  162.      7a-2z  +  8  =  33. 

160.  3a?-5  =  19.  163.      5x  -  3x  +  21  =  x  +  34. 

161.  5cc  =  x  +  36.  164.    lla  +  1  =  a;  +  91. 

165.  One  of  two  numbers  is  3  more  than  twice  the  other. 
Their  sum  is  15.     What  are  the  numbers  ? 

Let  x  =  the  less  number, 

then  2  x  +  3  =  the  greater  number, 

then  x  +  2  x  +  3  =  15. 

166.  There  are  two  numbers  whose  sum  is  17.     The  greater 
is  2  more  than  4  times  the  less.     Find  the  numbers. 

167.  In  the   broken  line  ABC,  BC 
represents  a  distance  which  is  7  ft.  more 
than  twice  AB.     If  the  entire  line  rep- 
resents 31  ft.,  how  much  does  AB  rep- 
resent?    BO? 

168.  Each  of  the  equal  sides  of  an 
isosceles  triangle  is  4  ft.  longer  than 

the  base.     The  perimeter  is  29  ft.     How  long  is  each  side? 
Represent. 

169.    Fred  is  seven  years  older  than  his  brother,  and  the 
sum  of  their  ages  is  23  years.     How  old  is  each  ? 


310  LITERAL  QUANTITIES 

170.  CLASS  EXERCISE.     may  think  of  the  ages  of  two 

persons,  and  tell  the  class  the  sum  of  those  ages  and  the  dif- 
ference between  them.     The  class  may  find  the  ages. 

171.  Mr.  Lee  has  a  watch  which  is  worth  $20  more  than 
the  chain.     The  watch  and  chain  together  are  worth   $50. 
How  much  is  each  worth  ? 

172.  Two  boys  bought  a  skiff  for  $  8.     The  older  boy  gave 
$  2  more  than  the  younger.     How  much  did  each  give  ? 

173.  A  bootblack   earned   30^  more  on  Tuesday  than  on 
Monday.     His  earnings  for  the  two  days  were  $  1.70.     How 
much  did  he  earn  on  each  of  these  days  ? 

174.  A  farmer,  who  had  100  acres  of  corn  and  wheat,  had  20 
acres  more  of  wheat  than  of  corn.     How  many  acres  of  corn 
had  he  ?     How  many  acres  of  wheat  ? 

175.  The  sum  of  two  numbers  is  72.     The  greater  is  8  more 
than  the  less.     Find  the  numbers. 

176.  The  sum  of  two  numbers  is  90,  and  the  greater  is  26 
more  than  twice  the  less.     Find  the  numbers. 

177.  A  traveled  north  from  the  Chicago  post  office,  and  B 
traveled   south   from   that   point.      When  they  were  50  ini. 
apart,  A  had  traveled  10  mi.   more  than  B.      How  far  was 
each  from  the  Chicago  post  office  ? 

178.  The  length  of  a  rectangular  lot  is  70  ft.  more  than  its 
width.     Its  perimeter  is  220  ft.     Represent  and  find  the  length 
and  the  area. 

179.  Edwin  went  fishing.     If  he  had  caught  10  times  as 
many  fish  as  he  did  catch  and  40  fish  more,  he  would  have  had 
100  fish.     How  many  did  he  catch  ? 

180.  Find  three  numbers  such  that  the  first  is  10  more  than 
the  second,  the  second  is  5  more  than  the  third,  and  their  sum 
is  47. 


LITERAL   QUANTITIES  311 

Let  x  =  the  smallest  or  third  number, 

then  x    4-  6  =  the  second  number, 

and  x  4-  5  +  10  =  the  first  number. 


181.  The  perimeter  of  the  triangle  which 
ABC  represents  is  36  in.  The  side  AB  is  3 
in.  longer  than  the  side  BC,  and  the  side  AC 
is  3  in.  longer  than  the  side  AB.  Find  length 
of  each  side. 

Represent  the  following  : 

B       ^+  G         182.    In  the  scalene  triangle  ABC,  the  side 

AB  is  12  in.  longer  than  the  side  AC.  The 
side  AC  is  8  in.  longer  than  the  side  BC.  The  perimeter  is 
73  in.  Find  each  side. 

183.  The  side  XYof  the  triangle  XYZis  11  in.  longer  than 
the  side  YZ,  and  the  side  XZ  is  17  in.  longer  than  the  side  YZ. 
The  perimeter  is  88  in.     Find  each  side. 

184.  In  the  triangle  DEF,  the  side  DE  lacks  8  in.  of  being 
twice  as  long  as  the  side  EF.     The  side  DF  lacks  17  in.  of 
being  three  times  as  long  as  EF.     The  perimeter  is  65  in. 
Find  each  side. 

185.  4  times  a  certain  number  =  that  number  -f  21.     Find 
the  number. 

186.  Separate  27  into  two  parts  such  that  the  greater  is  9 
more  than  the  less. 

187.  There  were  four  brothers,  the  sum  of  whose  ages  was 
32  yr.     Each  boy  was  2  yr.  older  than  his  next  younger  brother. 
How  old  was  each  ? 

188.  If  3  yr.  were  subtracted  from  4  times  John's  age,  the 
remainder  would  equal  his  father's  age,  which  is  45  yr.     How 
old  is  John  ? 


312  LITERAL   QUANTITIES 

189.  32  boys  voted  for  the  president  of  their  club.     John 
received  6  more  votes  than  the  other  candidate.     How  many 
votes  did  each  candidate  receive  ? 

190.  Mr.  A  pays  $  13  more  in  taxes  than  Mr.  B.     Mr.  C 
pays  $  7  more  than  Mr.  B.     Mr.  D  pays  $  8  more  than  Mr.  C. 
They  all  pay  $  69.     How  much  does  each  pay  ? 

191.  An  importer  received  three  shipments  of  goods  from 
Germany.     The  duty  on  the  second  shipment  was  $  3000  more 
than  on  the  first,  and  the  duty  on  the  last  was  $  2500  more 
than  on  the  second.     The  duty  011  all  the  shipments  amounted 
to  $  9500.     How  much  was  paid  on  each  ? 

192.  In  a  game  of  football  the  successful  team  scored  3 
times  as  many  points  as  the  other.     The   difference   in  the 
scores  was  18.     What  was  the  score  of  each  team  ? 

193.  Make  a  problem  to  be  solved  by  equations. 

194.  The  profits  of  a  farm  in  3  yr.  were  f  2300.     The  profits 
for  the  second  year  were  $  100  more  than  for  the  first  year. 
The  profits  for  the  third  year  were  $  300  less  than  for  the 
second  year.     Find  the  profits  for  each  year. 

195.  Mr.  Eowe  gave  3  notes  to  a  collector,  who  collected 
$  7  more  on  the  second  note  than  on  the  first,  and  on  the  third 
$  3  less  than  on  the  second.     The  sum  of  the  collections  was 
$  40.     How  much  was  each  ? 

196.  The  senior  partner  in  a  firm  has  $  20,000  more  in 
the  business  than  the  junior  partner.     The  whole  capital  is 
$  80,000.     What  is  the  capital  of  each  partner  ? 

197.  Mr.  A  owes  Mrs.  B  $  21  more  than  he  owes  Mr.  C. 
Both  debts  amount  to  $  225.     How  much  is  his  debt  to  Mr.  C  ? 
If  he  pays  $  7  a  week  to  Mrs.  B,  in  how  many  weeks  will  he 
have  paid  his  debt  to  her  ? 


LITEKAL   QUANTITIES  313 

198.  A  chord  AB  divides  a  circumference  into  two  arcs, 
the  greater  of  which  is  30  ft.  longer  than  twice  the  less.  The 
circumference  is  120  ft.  How  long  is  each  arc  ? 

199.  Multiplying  all   the  terms   of  an 
equation   by  the   same  number  has  what 
effect    upon    the     equation?       Illustrate 
with  numbers. 

200,  The  expression  | means  -  of  x;  -^ 
2  3  o  o 

means  -  of  x.     If  you  multiply  both  terms 
FIG.  18.  3 

of  the  fraction  ^  by  3,  to  what  integral  expression  is  the  result 
equal  ?  Why  ?  3 

201.  Multiply  by  the  same  number  both  terms  of  the  equa- 
tion -  =  7  and  solve  the  equation. 

3 

202.  Multiplying  the  terms  of  a  fractional  equation  by  a 
quantity  that  causes  the  terms   to   become  integral  is  called 
clearing  the  equation  of  fractions.     How  did  you  clear  of  frac- 
tions the  equation  in  Ex.  201  ? 

203.  Solve  -  +  -  =  7. 

3       4 

Multiplying  all  the  terms  by  3  we  have  x  +  —  =  21.     Multiplying  all 

4 

the  terms  of  that  equation  by  4  we  have  4  x  +  3  x  =  84.  It  would  have 
been  a  shorter  process  to  multiply  all  the  terms  by  12  at  once  instead  of 
by  the  3  and  4  separately.  Hence  we  use  the  method  given  below. 

204.  Solve  Ex.  203  by  the  following  rule : 
To  clear  an  equation  of  fractions  — 

Multiply  each  term  of  the  equation  by  the  least  common  mul- 
tiple of  the  denominators. 

205.  Clear  of  fractions  the  equation  -  -f  -  =  5. 

6      4 

Multiplying  each  term  by  12,  the  1.  c.  m.  of  6  and  4,  we  have 

|x!2  =  2z       ^x!2  =  3z      5x12  =  60.     Hence,  2  x  +  3  x  =  60. 

o  •          4 


314 


LITERAL  QUANTITIES 


Solve : 

206.    ^  +  ^=24.      207.    -  +  -  = 
57  69 


209.     —  +^=19f  215. 

7        5 


210.    2z  +  ^4-^  =  43Tiir.  216. 


208.    ^f  +  ^  = 
3       15 

o  .     £  QC          r»  r* 


218.  4  = 


213. 


214. 


219.          _.:=     . 
4        7       14 


220.         _4:= 


of  it  =  24.     Find 


of  it  are 


3       6 

221.  One  fifth  of  a  certain  number  + 
the  number. 

222.  Find  a  number  such  that  if  $  of  it  and 
added  to  it,  the  sum  will  be  28. 

'A  223.    The  parts  of  the  broken 

line  ABCD  are  in  the  ratios  given 
c  in  Fig.   19.     How    long    is   each 

part  if  the  entire  line  is  30  in.? 
75  in.?     74-  in.? 

224.    A  circumference  which  is 
FIG-  19-  1  yd.  in  length  is  divided  into  2 

arcs,  one  of  which  is  %  of  the  other.     How  long  is  each  arc  ? 
Represent. 

225.  Find  a  number  such  that  if  15  is  subtracted  from  3 
times  the  number,  the  remainder  will  be  2£  times  the  original 
number. 

226.  The  perimeter  of  a  given  isosceles  triangle  is  286  ft., 
and  the  base  is  T4r  of  one  of  the  equal  sides.     Find  the  length 
of  its  sides. 


LITERAL  QUANTITIES  315 

227.  The  perimeter  of  a  rectangle  is  1254  ft.,  and  the  width 
is  j^-  of  the  length.     Find  width,  length,  and  area. 

228.  A  certain  number  4-  2£  times  itself  +  7  =  37.     Find  the 
number. 

229.  Draw  a  right-angled  scalene  triangle.     If  the  altitude 
were  1|-  times  the  base,  the  hypotenuse  2^  times  the  base,  and 
the  perimeter  40  in.,  how  long  would  each  side  be  ? 

230.  Separate  45  into  two  parts,  one  of  which  is  J  of  the 
other. 

Let  x  =  the  greater  number. 

231.  42  is  the  sum  of  two  numbers  whose  ratio  is  f .     What 
are  they  ? 

232.  Separate  90  into  two  parts  whose  ratio  is  -J. 

233.  CLASS  EXERCISE.     may  give  the  sum  of  two  num- 
bers whose  ratio  is  a  fraction,  and  the  class   may  find  the 
numbers. 

234.  How  would  you  divide  75  $  between  two  boys,  giving 
one  boy  %  as  much  as  the  other  ? 

235.  The  session  of  a  certain  school  is  4J-  hr.  a  day.     How 
many  hours  and  minutes  are  given  to  recitation  if  the  recitation 
periods  take  f  as  much  time  as  that  devoted  to  other  purposes  ? 

236.  There  are  three  numbers  whose  sum  is  108.     The  first 
is  f  of  the  second,  and  the  third  is  twice  the  first.     Find  the 
numbers. 

Let  x  =  the  second  number. 

237.  4-  of  a  certain  number  minus  -J-  of  it  =  2.     What  is  the 
number  ? 

238.  There  are  three  numbers  whose  sum  is  84.    The  second 
number  is  1^-  times  the  first,  and  the  third  number  is  ^  of  the 
second.     Find  each  number. 

239.  The  profits  of  a  business  during  its  second  year  were 
1£  times  the  profits  during  its  first  year,  and  the  profits  for  the 


316  LITERAL   QUANTITIES 

third  year  were  1-^-  times  those  for  the  second  year.     The  profits 
for  the  3  yr.  were  $  8250.     Find  the  profits  for  each  year. 

240.  Draw  a  trapezoid  making  the  upper  base  £  the  lower 
base.     If  one  of  the  non-parallel  sides  is   J  of  the  lower  base, 
the  other  non-parallel  side  -f^  of  the  lower  base,  and  the  per- 
imeter of  the  trapezoid  is  54  in.,  how  long  is  each  side  ? 

241.  How  long  is  each  side  of  a  rhomboid  whose  perimeter 
is  14  ft.  6  in.  and  whose  short  sides  are  each  £  as  long  as  a 
long  side  ? 

MISCELLANEOUS    EXERCISES 

1.  7%  of  7%  of  $825  =  ? 

2.  A  lawyer  collected  $  1275  for  a  client.      He  charged 
10%  for  collecting.     He  gave  60%  of  his  fee  to  his  wife.     How 
much  money  was  received  by  the  client  ?     By  the  lawyer  ?     By 
his  wife  ? 

3.  Mr.  and  Mrs.  Shaw  and  two  children  took  a  trip  on  a 
lake  steamer.    The  fare  was  $  9.00,  children  half  price.     Meals 
on  the  steamer  were  $  1.00  each.      The  family  took  supper, 
breakfast,  and  dinner  on  board,  and   paid  $  5.00  for  a  state- 
room.    What  was  the  cost  of  the  trip  ? 

4.  Find  the  g.  c.  d.  of  the  first  composite  odd  number  after 
39  and  the  first  composite  odd  number  after  57. 

5.  How  much   is  gained  by  buying  a  $  500  bond  at  105, 
keeping  it  until  2  yr.  interest  at  3%  has  been  received,  and 
selling  it  at  109  ? 

6.  In  a  certain  city,  the  highest  temperature  in  July  was 
100°.     The  highest  temperature  in  December  was  75°.      The 
difference  in  temperature  was  what  per  cent  of   the  highest 
temperature  in  July  ?     In  December  ? 

7.  Solve  9  a  -  25 -f  3x  =  7x-5. 

8.  Solve  2lx-20  =  7x-  15^  +  67. 

9.  Three  times  a  certain  number  equals  148  minus  the 
number.     What  is  the  number  ? 


MISCELLANEOUS  EXERCISES  317 

10.  What  number  doubled  and  increased  by  4  equals  188  ? 

11.  If  Mary  were  15  years  older  than  twice  her  present  age, 
she  would  be  as  old  as  her  cousin,  who  is  37  years  old.     How 
old  is  Mary  ? 

12.  Find  a  number  such  that  the  difference  between  ^  of  it 
and  -j-  of  it  is  2. 

13.  The  perimeter  of  a  rhomboid  is  70  dm.      Its  long  sides 
are  each  5  dm.  longer  than  the  sum  of  its  short  sides.     Find 
the  length  of  each  side. 

14.  CLASS  EXERCISE.     may  write  an  equation  having  90 

for  its  second  member  and  a  prime  number  for  the  coefficient 
of  the  unknown  quantity.     The  class  may  solve  the  equation. 

15.  Solve  .6^  =  120. 

Clear  the  equation  of  fractions  by  multiplying  each  term  by  the  denom- 
inator of  the  decimal  .6. 

16.  Solve  .06  x  =  24,     .03  x  =  $  240,     8  %  of  x  =  32.64. 

17.  Solve  .7  x  =  2800,     .016^  =  32,     4%  a  =  48. 

18.  Solve  .9  x  =  540,     11  %  x  =  33,     .012  x  =  720. 

19.  An  agent  who  charged  7%  for  collecting  a  sum  of  money, 
received  $  210  as  his  commission.     How  much  did  he  collect  ? 

Let  x  equal  the  number  of  dollars  collected,  then  .07  x  equal  210. 

20.  What  amount  must  be  collected  that  the  fee  for  collecting 
it  may  be  $70  when  the  rate  is  5%  ?    7%  ?    2%  ?    10%?    S%? 

21.  Express  in  terms  of  x  the  interest  of  $  x  for  1  yr.  6  mo. 
at  6%. 

The  interest  of  $  1  for  1  yr.  6  mo.  at  6  %  is  9  ^.  The  interest  of  $  x  is 
x  times  9^  or  9x^. 

22.  Express  in  terms  of  x  the  interest  of  $  x  at  6%  for  2  yr. 
6  mo.  6  da.     For  5  yr.  8  mo.  12  da.     For  7  yr.  10  mo.  24  da. 

23.  What  principal  will  gain  $  157.50  in  3  yr.  6  mo.  at  6%  ? 

Let  x  =  the  number  of  dollars  in  the  principal.  $  1  in  3  yr.  6  mo.  at 
6%  will  gain  §.21.  x  dollars  will  gain  x  times  $.21  or  $.21«.  Then 
$.  21  x  =  $157.50. 


318  LITERAL  QUANTITIES 

24.  By  similar  reasoning  find  the  principal  which  will  gain 
$  19.75  in  2  yr.  3  mo.  at  6%.  When  you  have  found  it,  see  if 
the  interest  upon  it  at  6%  for  2  yr.  3  mo.  is  $  19.75. 

SUGGESTION  TO  TEACHER.  Pupils  should  prove  these  problems  until 
they  realize  that  each  of  them  is  merely  a  reversed  case  of  the  ordinary 
problem  in  which  the  interest  is  required  to  be  found. 

Find  the  principal  which  will  gain : 

25.  $240  in  3  yr.  at  5%. 

26.  $  360  in  4  yr.  6  mo.  at  6%. 

27.  $780  in  5  yr.  at  8%. 

28.  $  175  in  6  yr.  3  mo.  at  4%. 

29.  $  200  in  3  yr.  2  mo.  15  da.  at  8%. 

30.  $  250  in  2  yr.  8  mo.  at  6%. 

31.  In  what  time  will  $  500  gain  $  34  at  6%  ? 

Let  x  =  the  number  of  years.  The  interest  of  $  500  at  6  %  for  1  yr.  is 
$  30.  For  x  yr.  the  interest  will  be  x  times  $  30  or  30  x  dollars.  Hence 
30  x  —  34  and  x  ^  1^  yr.  or  1  yr.  1  mo.  18  da. 

In  what  time  will : 

32.  $560  gain  $106.40  at  8%  ? 

33.  $  750  gain  $  192  at  6%  ? 

34.  $  187.50  gain  $  37.50  at  5%  ? 

35.  $  65  gain  $  2.60  at  6%  ? 

36.  $216  gain  $122.22  at  10%? 

37.  At  what  per  cent  will  $  400  gain  $  35  in  2£  yr. 

Let  x  =  the  number  of  per  cent.  The  interest  of  $  400  for  2£  yr.  at 
1  %  is  $  10.  At  x  %  the  interest  will  be  x  times  $  10  or  10  x  dollars. 
Hence  10  x  =  35  and  x  =  3  \  %. 

At  what  per  cent  will : 

38.  $  700  earn  $  63  in  2  yr.  3  mo.  ? 

39.  $  600  earn  $  45  in  1  yr.  6  mo.  ? 


MISCELLANEOUS  EXERCISES  319 

40.  $  225  earn  $  49.50  in  2  yr.  9  mo.  ? 

41.  $  500  earn  $  105  in  7  yr.  ? 

42.  $  600  earn  $  125  in  8  yr.  4  mo.  ? 

43.  What  principal  will  amount  to  $  532  in  3  yr.  8  mo.  at  9% . 

Let  x  =  the  number  of  dollars  in  the  principal.    The  amount  of  $  1  for 
3  yr.  8  mo.  at  9  %  =  $  1.33;  the  amount  of  x  dollars  =  1.33  x  dollars. 

44.  What  principal  will  amount  to  $  94.50  in  2  yr.  6  mo. 
at  5%  ? 

45.  $  155  in  3  yr.  4  mo.  at  6%  ? 

46.  $  85  in  1  yr.  8  mo.  at  8%  ? 

47.  $  168.36  in  2  yr.  5  mo.  at  7%  ? 

Find  the  missing  term  in  the  following : 

Prin.                    Rate                     Time  Int. 

48.  $600                 5%                 2£yr.  x 

49.  $400                x                    3   yr.  $48 

50.  x  7%  l£yr.  $31.50 

51.  $500  6%  x  $50 

52.  A  man  wishes  to  set  aside  a  sum  of  money,  the  interest 
of  which  will  furnish  his  daughter  a  yearly  income  of  $  1000. 
If  6%  can  be  obtained  for  it,  how  much  shall  he  invest  for  her? 
How  much  would  be  necessary  to  invest  if  only  3-|-%  could  be 
obtained  for  it  ? 

53.  Which  is  the  greater  price  for  an  article,  $  100  cash  or 
$  108  due  in  1  yr.,  without  interest,  when  the  customary  interest 
is  8%.     Why? 

54.  The  Present  Worth  of  a  sum  of  money  due  at  a  given 
time  is  that  smaller  sum  of  money  which,  when  put  at  interest 
at  the  usual  rate,  will  amount  to  the  given  sum  in  the  given 
time. 

What   is  the  present  worth  of  $  770  due  in  2  yr.,   when 
money  is  worth  5%  ? 

The  above  question  really  asks,  what  sum  put  at  interest  at  5%  will  in 
2  yr.  amount  to  $  770  ? 


320  LITERAL   QUANTITIES 

55.  When  money  is  worth  6%,  what  is  the  present  worth  of 
$224,  due2yr.  hence? 

When  you  have  found  the  present  worth,  prove  your  work  by  comput- 
ing the  interest  upon  it  for  the  given  time  and  rate  to  see  whether  the 
principal  and  interest  will  amount  to  $  224. 

56.  The   difference    between   the   present   worth   and    the 
amount  due  at  maturity  is  called  the  True  Discount. 

Supposing  money  to  be  worth  6%,  find  the  present  worth 
and  the  true  discount  of  $  4720,  due  3  yr.  hence. 

True  discount  must  be  distinguished  from  bank  discount,  which  is 
merely  the  simple  interest  on  the  face  of  the  note. 

Find  present  worth  and  true  discount  of :    , 

57.  $199.80,  duel  yr.  10  mo.  hence. 

58.  $307.50,  due  5  mo.  hence. 

59.  $  143.75,  due  2  yr.  6  mo.  hence. 

60.  There  are  two  ways  of  finding  the  true  discount  after 
the  present  worth  is  known.     What  are  they  ? 

61.  CLASS  EXERCISE.   may  think  of  a  sum  of  money  and 

find  how  much  it  will  be  worth  at  a  given  future  time,  the  cus- 
tomary rate  of  interest  being  — °/0.    He  may  report  to  the  class 
the  amount  of  that  sum,  and  the  time  and  rate,  and  the  class 
may  find  the  original  sum. 

Find  the  difference  between  the  bank  discount  and  the  true 
discount  of  the  following,  the  rate  of  interest  being  4% : 

62.  $856.00,  due  in  1  yr.  9  mo. 

63.  $817.50,  due  in  2  yr.  3  mo. 

64.  $712.50,  due  in  3  yr.  6  mo. 

65.  $  626.00,  due  in  1  yr.  1  mo. 

66.  $  987.00,  due  in  2  yr.  5  mo. 

67.  Kesolve  18  into  two  factors,  one  of  which  is  a  perfect 
square. 


MISCELLANEOUS   EXERCISES  321 

68.  Resolve  108  into  two  factors,  one  of  which  is  the  largest 
possible  square. 

69.  Find  the  area  of  a  right  triangle  whose  base  is  20  in. 
and  altitude  1  ft. 

70.  Find  the  area  of  a  trapezoid  whose  upper  base  is  10  in., 
lower  base  14  in.,  and  altitude  5  in. 

71.  Find  the  interest  of  $1000  for  3  yr.  at  6%. 

72.  If  Mr.  Brown  puts  $1000  at  interest  at  6%,  how  much 
interest  will  be  due  him  at  the  end  of  one  year  ?     If,  instead 
of  collecting  this  interest,  he  adds  it  to  the  principal,  and  loans 
the  $1060  for  a  second  year  at  6%,  what  will  be  the  second 
year's  interest?      If  this  second  year's  interest,   $63.60,  is 
added  to  the  second  principal,  $  1060,  and  the  sum,  $  1123.60, 
is  put  at  interest  for  the  third  year,  what  will  be  the  amount 
at  the  end  of  that  year  ? 

73.  How  much  greater  is  that  amount  than  the  original 
$  1000  ?     That  increase  is  the  Compound  Interest.     How  much 
greater  is  the  compound  interest  than  the  ordinary  simple 
interest  of  $  1000  for  3  yr.  at  6%? 

74.  Find  by  the  method  shown  in  Exs.  72  and  73  the  com- 
pound interest  of  $  1000  for  4  yr.  at  10%. 

75.  Money  put  at  compound  interest  gains  more  rapidly  as 
the  number  of  years  increases.     Can  you  see  why  ? 

76.  One  dollar  put  at  compound  interest  at  7%  will  amount 
in  25  yr.  to  $  5.427.     To  how  much  will  $  2000  amount  in  that 
time  at  that  rate  of  compound  interest  ?     How  much  of  that 
sum  is  interest  ?     Find  the  simple  interest  of  $  2000  for  25  yr. 
at  7%,  and  find  how  much  less  it  is  than  the  compound  interest. 

77.  On  his  eighteenth  birthday  John  Smith  deposited  $  100 
in  a  savings  bank,  which  paid  4%  compound  interest.    He  did 
the  same  on  his  next  two  birthdays.     How  much  had  he  to 
his  credit  in  that  bank  on  his  twenty-first  birthday  ? 

HORN.    GRAM.    SCII.    AR. — 21 


CHAPTER  X 

INVOLUTION  AND  EVOLUTION 

1.  What  is  meant  by  the  power  of  a  number  ?     Illustrate. 

2.  Raise  to  the  3d  power  19.     £.     .3.     2£.     1.2.     .000007. 

3.  The  process  of  raising  a  quantity  to  a  higher  power  is 
called  Involution. 

Involve  74.     .25.     (4i)3.     .63.     (l.l)4.     (£)2. 

4.  93:35  =  ?     83:27  =  ?     64:47  =  ? 

5.  When  a  =  2,  how  much  is  3  a5  ? 

6.  Give  quickly  the  squares  of  the  first  12  numbers. 

7.  Give  the  squares  of  20.     30.     40.     80.     70. 

8.  Give  quickly  the  cubes  of  the  first  12  numbers. 

9.  Give  the  cubes  of  20.     40.     60.     80.     50. 

10.  Name  a  perfect  square  that  is  a  factor  of  72.     Of  50. 

11.  Name  a  perfect  cube  that  is  a  factor  of  250.     Of  24. 

12.  What  is  meant  by  the  root  of  a  number  ?     Illustrate. 

13.  The  process  of  finding  any  root  of  a  given  quantity  is 
called  Evolution.     It  is  the  exact  opposite  of  involution. 

What  is  the  square  root  of  121  ?     Of  144  ? 

14.  Draw  a  square,  and  illustrate  the  following  statement : 

The  number  of  units  of  length  in  one  side  of  a  square  is  the 
square  root  of  the  number  of  corresponding  units  of  square 
measure  in  the  area  of  the  square. 

322 


INVOLUTION  AND  EVOLUTION 


323 


15.  Copy  Fig.  1,  making  ABCD  a  4-inch  square,  and  AE 
and  CO  each  2  in.  How  many  square 
inches  in  the  square,  ABCD?  In 
HFKB?  In  the  rectangle,  BKCG? 
In  the  rectangle  EHBA?  How  many 
square  inches  in  all  the  additions  to  the 
square  ABCD  ? 

16.    How  long  is  one  side  of  a  square 
which  contains  4  sq.  ft.     Represent  it. 
Draw   the    additions    which    would    be 
needed  to  make  the  figure  represent  a  square  yard. 

17.  Draw  a  square  which  contains  81  sq.  cm.     Add  squares 
to  it  until  it  is  a  square  decimeter.     How  many  square  centi- 
meters are  there  in  the  two  rectangles  and  little  square  which 
were  added  ? 

18.  How  many  square  inches  are  there  in  the  two  rectangles 
and  the  small  square  which,  when  added  to  a  5-inch  square, 
will  change  it  to  an  8-inch  square  ? 

19.  How  many  square  inches  are  there  in  the  two  rectangles 
and  small  square  which,  when  added  to  a  10-inch  square,  will 
change  it  to  a  14-inch  square  ? 


additions  on  two  sides.  How  wide 
is  each  addition  ?  How  many 
square  "inches  are  there  in  the 
sum  of  all  the  additions? 

21.  If  the  square  ABCD  in 
Fig.  2  contains  100  sq.  in.,  how 
long  will  the  line  DC  be  ?  If  the 
sum  of  all  the  additions  is  69  sq. 
in.,  how  long  will  the  line  CG 
be? 


20.    A  10-inch  square 

gr                           HI 

n 

FIG.  2. 


324 


INVOLUTION  AND  EVOLUTION 


10x10 


H 

B        K 


10x10 


FIG.  3. 


FIG.  4. 


SOLUTION.     CG-  in  Fig.  4  represents  the  width  of  the  additions  which 
will  change  the  square  ABGD  (Fig.  3)  to  the  square  EFGD  (Fig.  2). 


E 


n  K 


1                 10                 B  B                10 
FIG.  5. 

G 

"When  the  two  rectangles  in  Fig.  4  are  placed  side  by  side  as  in  Fig.  5, 
they  form  a  rectangle  whose  length  is  10  ft.  +  10  ft.,  or  20  ft.  We  know 
that  its  area  plus  the  area  of  the  square  needed  to  complete  Fig.  4  is 
69  sq.  ft.  We  wish  to  know  its  width.  Dividing  69,  the  number  of  square 
feet  in  all  the  additions,  by  20,  the  number  of  feet  in  the  base  of  the  two 
rectangles,  we  have  the  quotient  3,  which  shows  that  the  probable  width 
of  the  rectangles  is  3  ft.,  and  that  a  side  of  the  small  square  needed  to 
complete  Fig.  4  is  3  ft. 


H  K 


G  JIS   F 


A.                 10                 B  B                10                C  J5  8    K 
-     FIG.  6. 

When  the  small  square  is  added  to  the  sum  of  the  rectangles  as  in 
Fig.  6,  a  rectangle  is  formed  whose  length  is  10  ft.  +  10  ft.  -f  3  ft.,  or 
23  ft.  Assuming  that  its  width  is  3  ft.,  its  area  is  69  sq.  ft.,  which  exactly 
equals  all  the  square  feet  which  were  to  be  added  to  the  10-inch  square. 


INVOLUTION  AND   EVOLUTION  325 

22.  How  many  square  feet  would  be  represented  by  Fig.  4 
if  the  square  were  completed  ?     How  long  would  one  side  of 
it  be  ?     What,  then,  is  the  square  root  of  169  ? 

23.  A  lot  30  ft.  square  was  increased  by  the  addition  of 
61  sq.  ft.     The  additions  were  made  on  two  sides,  and  in  such' 
a  way  that  the  lot  when  increased  was  also  in  the  form  of  a 
square.    Eepresent  and  make  additions  as  in  the  solution  of  Ex. 
21.    How  wide  was  each  addition  ?     How  long  was  a  side  of  the 
square  after  the  additions  were  made  ?     How  many  square  feet 
were  in  the  completed  square  ?    What  is  the  square  root  of  961  ? 

24.  A  square  contains  100  sq.  in.     If  96  sq.  in.  are  added 
to  it,  as  in  Fig.  2,  how  wide  will  the  addition  be  ?     How  long 
will  one  side  of  the  whole  square  be?     What  is  the  square t 
root  of  196  ? 

25.  How  wide  must  be  the  additions  that  will  change  a 
square  containing  400  sq.  in.  to  a  square  containing  441  sq.  in  ? 
What  is  the  length  of  one  side  of  a  square  containing  441  sq.  in.  ? 
What  is  the  square  root  of  441  ? 

26.  If  you  had  576  sq.  ft.  of  boards  to  be  arranged  in  the 
form  of  a  square,  how  long  would  one  side  of  the  square  be  ? 

First  make  a  square  containing  400  sq.  ft.,  then  apply  the  remaining 
176  sq.  ft.  on  two  sides. 

27.  Find  in  the  same  way  the  side  of  a  square  containing 
625  sq.  in.     What  is  the  square  root  of  625  ? 

28.  12  =  1.     92  =  81.     102  =  100.     992  =  9801. 

How  many  figures  are  there  in  the  expression  of  the  square 
of  a  number  less  than  10  ?  In  the  expression  of  the  square  of 
a  number  greater  than  9  and  less  than  100  ? 

To  find  how  many  figures  there  are  in  the  integral  root  of  a  given 
number  separate  the  number  into  groups  of  two  figures  each  by  placing 
an  arc  over  the  units'  and  tens'  figures,  and  also  over  each  succeeding  group 
or  part  of  a  group,  as  12544.  The  number  of  arcs  equals  the  number  of 
figures  in  the  root. 


326  INVOLUTION  AND  EVOLUTION 

29.  How  many  figures  are  there  in  the  integral  root  of  a 
number  whose  expression  takes  three  places  ?     Six  places  ? 
Nine  places  ? 

30.  When   t  =  40   and  u  =  5,  how  much   is   (t  +  u)2  ? 


31.  When   £  =  30   and   u  =  2,  how  much  is   (t  +  u)2? 
Z2  +  2  tu  +  u2? 

32.  Find  the  square  of  25  in  terms  of  its  tens  and  units. 

25  =  2  tens  and  5  units. 

20  +  5          Beginning  at  the  right,  and  multiplying  units 

_  20  +  5      by  units,  we  have  5x5,  expressed  5'2.     Multiply- 

(20  x  5)  +  52     ing  tens  by  units  we  have  (20  x  5).     Multiplying 

202  +  (20  x  5)  5  units  by  20  we  have  the  equivalent  of  another 

202  +  2  (20  x  5)  +  52     (20  x  5),  which  is  written  under  the  first.   20  x  20 

is  expressed  202.     The  sum  of  all  these  products 

is  202  +  2  times  (20  x  5)  +  52. 

33.  Is  there  any  difference  between  the  value  of  252  and  of 
202  +  2(20  x  5)  +  52? 

34.  Illustrate  the  following  principle: 

The  square  of  any  number  consisting  of  tens  and  units  is  equal 
to  the  square  of  the  tens  plus  twice  the  product  of  the  tens  and 
units  plus  the  square  of  the  units. 

This  principle  may  be  expressed  by  the  following  formula: 
(t  +-  u*)2  =  t2  +-  2  tu  +-  u2.  By  the  use  of  this  formula  we  may 
readily  find  the  square  root  of  any  number. 

35.  Find  the  square  root  of  5329. 

^^  Beginning  at  the  right  and  separating  5329 

&  +  2  tu  +  w2  =  5329  |73      into  groups  of  two  figures  each,  we  find  that 

t2  =  4900  the  root  will  consist  of  tens  and  units.     We 

2  1  +  u  =  143)  429  find  that  4900   is  the  largest   square   of   a 

u  (2  1  +  w)  =    429  multiple  of  ten  that  5300  includes.     Here, 

4900  =  £2,  and  t  =  70.     We  write  the  7  as  the 

tens'  figure  of  the  root.  Subtracting  J2,  or  4900,  from  5329,  we  have 
remaining  429,  which  equals  2  tu  +  w2.  As  t  equals  70,  2  t  equals  140. 
We  wish  to  find  the  value  of  u.  We  use  140  as  a  trial  divisor,  and 
place  the  quotient  3  as  the  unit  figure  of  the  root.  We  add  the  3  to 


INVOLUTION  AND  EVOLUTION          327 

140  to  find  the  true  divisor.  Multiplying  the  true  divisor  143  by  the 
quotient  3  and  subtracting  the  product  from  429,  we  find  that  there  is  no 
remainder.  Hence  the  number  5329  is  a  perfect  square,  and  73  is  the 
square  root. 

NOTE  TO  TEACHER.  It  is  suggested  that  when  this  explanation  is  fully 
understood,  the  process  be  shortened  by  adopting  the  following  method 
for  the  work,  after  the  tens'  figure  of  the  root  has  been  found,  and  its 
square  subtracted  from  the  left  hand  period. 

"Double  the  root  already  found  and  place  the  result  as  a  trial  divisor. 
Cover  the  right-hand  figure  of  the  dividend,  and  see  how  many  times  the 
trial  divisor  is  contained  in  what  remains.  Place  the  quotient  beside  the 
tens'  figure  as  the  next  figure  of  the  root,  and  annex  it  also  to  the  trial 
divisor,  thus  making  the  true  divisor.  Multiply  the  true  divisor  by  the 
quotient  and  subtract  the  product  from  the  dividend,"  etc. 

36.  Find  the  square  root  of  1225. 

Extract  the  square  root  of  the  following : 

37.  625  43.    1764  49.  2809 

38.  484  44.    9216  50.  2025 

39.  289  45.    6889  51.  1089 

40.  1681  46.    2601  52.    6561 

41.  4356  47.    8649  53.    1156 

42.  2209  48.    5184  54.   5041 

55.  Find  the  square  root  of  324. 

Although  2  is  contained  in  22  eleven  times,  yet  nothing  larger  than  9 
can  be  used  for  a  root  figure. 

56.  V792i  =  ?         V2401  =  ?         V~348l  =  ? 

57.  Of  what  number  is  15  the  square  root  ?     20  ? 

58.  Give  a  perfect  square  whose  root  is  a  prime  number. 

59.  CLASS  EXERCISE.     may  give  to  the  class  the  square 

of  a  number  of  two  places.     The  class  may  find  the  number. 

60.  Find  the  square  root  of  12544. 

There  are  three  figures  in  the  root.  After  finding  the  hundreds'  and 
tens'  figures  double  the  root  already  found  and  proceed  as  before. 


328  INVOLUTION  AND   EVOLUTION 

Find  the  square  root  of  the  following  numbers : 

61.  245,025                     66.    249,001                     71.  511,225 

62.  375,769                     67.    546,121                     72.  811,801 

63.  784,996                     68.      56,169                    73.  674,041 

64.  776,161                    69.     23,409                    74.  70,756 

65.  938,961                    70.   529,984                    75.  119,716 

76.  How  long  is  a  side  of  a  square  whose  area  is  1681  sq.  in.  ? 
961  sq.  cm.  ?     729  sq.  ft.  ?     1024  sq.  m.  ? 

77.  How  long  is  the  perimeter  of  a  square  whose  area  is 
1089  sq.  in.  ?     8836  sq.  ft.  ?     5476  sq.  cm.  ?     3969  sq.  dm.  ? 

78.  A  field  contains  3600  sq.  rd.     How  much  will  it  cost  to 
fence  it  at  80  $  a  rod  ? 

79.  At  $  .60  a  rod  how  much  will  it  cost  to  fence  a  square 
field  containing  1849  sq.  rd  ?     1369  sq.  rd  ? 

80.  Find  the  square  root  of  -J.     Of  -f^. 

81.  Find  the  square  root  of :       -fff        ¥\9¥.        yf^.        T8^. 

82.  Keduce  1-j-  to  an  improper  fraction  and  find  its  square 
root. 

83.  Find  the  square  root  of :    2J.      4ff      3^.      2^.      6J|. 

84.  CLASS  EXERCISE.    may  give  to  the  class  the  square 

of  a  small  mixed  number.     The  class  may  find  the  number. 

85.  Find  the  square  root  of  the  decimal  .000625. 

In  dividing  a  decimal  into  groups,  begin  at  the  decimal  point  and  group 

to  the  right  as  .000625.     There  will  be  as  many  decimal  places  in  the 
root  as  there  are  groups  in  the  original  decimal. 

86.  Find  the  square  root  of:   .0049.    .0081.    .0025.    .000036. 

87.  Find  the  square  root  of:   .000529.        .000121.        29.16. 

88.  Give  one  of  the  two  equal  numbers  whose  product  is: 
50.41.        60.84.        94.2841.    "    38.3161. 

89.  Which  is  greater,  124  or  124.0000  ? 

90.  Find  the  approximate  square  root  of  124  to  two  places 
of  decimals. 

Annex  naughts  to  124. 


INVOLUTION  AND   EVOLUTION 


329 


91.  Find  to   two  places  of  decimals  the  square   root  of: 
27.         24.         85.        1.35.        2.7.        3.81.        4.09. 

92.  Find  to  one  decimal  place  the  length  of  the  side  of  a 
square  that  contains  45  sq.  in.         75  sq.  in.         95  sq.  in. 

93.  Find  value  of  x,  when  y?  =  .014641.     When  tf  =  .3969. 

94.  Draw  a  right  triangle  mak- 
ing the  base  3  in.  and  altitude  4 
in.     If  your  drawing  is  correct  how 
long  will  the  hypotenuse  be  ? 

95.  Construct  a  square  on  each 
side  of  the  triangle  as  in  Fig.  7  and 
compare  the  square  of  the  hypote- 
nuse with  the  sum  of  the  squares  of 

YIG.  7.  the  shorter  sides. 

96.  Geometry  shows  the  reasons  for  the  following  fact, 
called  the  "Pythagorean  Theorem"  after  Pythagoras,  the 
Greek  philosopher  who  gave  it  to  the  world. 

In  a  right  triangle  the  square  of  the  hypotenuse  is  equal  to 
the  sum  of  the  squares  of  the  other  two  sides. 

How  long  is  the  hypotenuse  of  a  right  triangle  whose  alti- 
tude is  12  in.  and  base  5  in.  ? 

Find  the  hypotenuse  of  each  of  the  following  triangles  : 

Base.        Altitude.  Base.        Altitude. 

97.  7  24  100.  15  20 

98.  8  15  101.  60  11 

99.  40  9  102.  24  45 

103.  A  traveled  east  120  miles  and  B  south  160  miles  from 
the  same  point.     How  far  apart  were  they  then  ? 

104.  How  long  is  the  diagonal  of  a  rectangle  which  is  24  in. 
long  and  10  in.  wide  ? 

105.  How- long  is  the  longest   straight  line   that  can  be 
drawn  on  a  blackboard  12  ft.  long  and  5  ft.  .wide  ?: 


330          INVOLUTION  AND  EVOLUTION 

106.  If  your  schoolroom  were  36  ft.  long  and  27  ft.  wide, 
what  would  be  the  length  of  the  longest  straight  line  that  could 
be  drawn  upon  the  floor  ? 

107.  Draw  a  vertical  line  AB  8  in.  long, 
and  mark  the  middle  point  E.     Draw  CD,  a 
horizontal  line,  6  in.  long,  whose  middle  point 
is  also   E.     Draw  AD,  DB,  BC,  and  CA. 

~~D    What   kind   of  a  figure    do  they   outline? 
How  long  is  its  perimeter? 

108.  The  diagonals  of  every  rhombus  bi- 
B                 sect   each  other   at  right   angles.      If    the 

diagonals  of  the  rhombus  you  have  drawn 
were  16  in.,  and  12  in.,  how  long  would  a  side  of  the  rhombus 
be? 

109.  How  long  is  the  perimeter  of  a  rhombus  whose  long 
diagonal  is  2  ft.,  and  short  diagonal  1J  ft.  ? 

110.  A  gate  8  ft.  long  and  6  ft.  wide  has  a  diagonal  cross- 
bar.   How  long  a  piece  of  wood  was  required  for  the  crossbar  ? 

111.  A  window  is  12  ft.  from  the  ground.     How  long  a 
ladder  is  required  to  reach  it  if  the  foot  of  the  ladder  is  16  ft. 
from  the  foundation  of  the  house  ? 

112.  John's  kite  is  caught  in  the  top  of  a  tree  69  ft.  high, 
and  John,  who  is  standing  156  ft.  from  the  tree,  holds  the  kite 
by  the  end  of  the  string.     His  hand  is  4  ft.  from  the  ground. 
Represent  the  conditions  of  this  problem  and  find  the  length  of 
the  string. 

113.  How  long  is  the  perimeter  of  a  right  triangle  whose 
base  is  21  ft.  and  altitude  72  ft.  ? 

114.  In  a  right  triangle  whose  base  is  80  cm.  and  altitude 
18  cm.  the  hypotenuse  is  how  much  less  than  the  sum  of  the 
other  two  sides  ? 

115.  What  distance  is  saved  by  walking  diagonally  across 
a  vacant  lot  300  ft.  long  and  160  ft.  wide  instead  of  walking 
along  two  sides  of  it  ? 


INVOLUTION  AND  EVOLUTION  331 

116.  By  taking  the  diagonal  path  across  a  rectangular  lot 
instead  of  walking  between  the  same  points  along  the  edge  of 
the  lot,  what  fractional  part  of  the  distance  is  saved  when  the 
lot  is  42  ft.  long  and  40  ft.  wide  ?     75  ft.  long,  40  ft.  wide  ? 
80  ft.  long,  60  ft.  wide  ? 

117.  Since  the  square  of  the  hypotenuse  equals  the  sum  of 
the  squares  of  the  shorter  sides,  the  square  of  either  short  side 
is  equal  to  the  square  of  the  hypotenuse  minus  the  square  of 
the  third  side.     Let  b  stand  for  base,  a  for  altitude,  and  h  for 
hypotenuse.     Show  how  the  following  equations  are  derived : 

62  +  a2  =  h2 
62  =  7i2  -  a2 
a2  =  7*2  -  62 

Use  the  equations  of  Ex.  117  in  finding  the  missing  side  of 
triangles  which  have  the  following  measurements. 

Base.          Alt.  Hyp.  Ease.         Alt.          Hyp. 

118.  18  x          30  121.  48  x  50 

119.  x          12          26  122.     x          72          75 

120.  7  12  a  123.     #  9  20 

124.  The  center  pole  of  a  round  tent  is  18  ft.  high.     A  rope 
stretches  from  a  point  3  ft.  from  the  top  of  the  pole  to  a  point 
on  the  ground  36  ft.  from  the  base  of  the  pole.     How  long  is 
the  rope  ? 

125.  Draw  a  right-angled  isosceles  triangle.     If  each  of  the 
equal  sides  were  9  in.,  how  long  would  the  hypotenuse  be  ? 

126.  How  long  is  the  diagonal  of  a  square  each  of  whose 
sides  is  10  ft.  ? 

127.  How  long  is  the  diagonal  of  a  square  whose  area  is 
64  sq.  ft.  ? 

128.  The  diagonal  of  a  square  is  15  in.,  how  long  is  one  side 
of  the  square  ? 

*2  +  X2  =  152 


332  INVOLUTION  AND   EVOLUTION 

129.  A  chimney  60  ft.  high,  casts  a  shadow  144  ft.  long. 
What  is  the  distance  from  the  end  of  the  shadow  to  the  top  of 
the  chimney  ? 

130.  Try  to  find  an  integer  whose  square  is  equal  to  the 
sum  of  the  squares  of  two  other  integers.     With  the  numbers 
make  a  problem  about  finding  one  of  the  sides  of  a  right 
triangle. 

The  following  is  an  easy  way  of  finding  three  integers,  the  square  of 
one  of  which  is  equal  to  the  sum  of  the  squares  of  the  other  two.  Take 
any  two  unequal  numbers.  The  sum  of  their  squares  will  represent  the 
hypotenuse  of  a  right  triangle,  the  difference  of  their  squares  one  of  its 
sides,  and  twice  their  product  the  other  side.  Taking  2  and  1, 

22+12  =  5,     22-12  =  3,     and2(2xl)=4. 

131.  Find  three  integers  which  may  represent  the  length  of 
the  three  sides  of  a  right  triangle,  by  combining  3  and  1  as  di- 
rected in  the  previous  note.     Combine  4  and  1,  3  and  2,  1  and 
1,  5  and  2.     Prove  your  work. 

In  the  following  problems,  when  the  square  root  is  not  exact,  extract 
the  root  to  one  decimal  place. 

132.  In  a  room  20  ft.  long,  15  ft.  wide  and  10  ft.  high,  a 
beetle  crossed  the  floor  diagonally  and  then  crawled  up  to  the 
ceiling.     How  far  did  he  travel  ?     What  was  his  shortest  dis- 
tance then  from  the  point  at  which  he  started  ? 

Below  are  given  the  dimensions  of  several  rooms.     Find  dis- 
tances from  an  upper  corner  to  the  opposite  lower  corner. 
Length.  Width.  Height. 

133.  24  ft.  18  ft.  9  ft.  . 

134.  17ft.  15ft.  10ft. 

135.  25ft.  24ft.  lift. 

136.  A  tank  8  ft.  long  and  6  ft.  wide  has  water  in  it  to  the 
depth  of  5  ft.     How  long  is  the  longest  straight  stick  that  can 
be  wholly  in  the  water  if  the  stick  is  pointed  at  both  ends  ? 


MISCELLANEOUS   EXERCISES  333 

MISCELLANEOUS   EXERCISES 

1.  A  party  of  excursionists,  consisting  of  9  adults  and  4 
children,   went   from   St.   Louis   to   Washington,  D.  C.,  and 
returned.     Eound  trip  tickets  were  $  25.45,  children  half  price. 
The  party  occupied  7  berths  in  the  sleeping  car  in  going,  and 
the  same  number  in  returning.      The  price  of  each  berth  was 
$4.75  each  way.     Meals  in  the  dining  car  cost  $1.00  each. 
The  party  left  St.  Louis  at  4  P.M.,  on  Monday,  and  arrived  in 
Washington  in  time  for  breakfast  Wednesday  morning.    During 
the  trip  the  whole  party  took  each  meal  in  the  dining  car. 
They  took  an  equal  number  of  meals  on  the  cars  on  their  return 
trip.     What  were  the  expenses  of  the  whole  party  on  the  trip? 

2.  Multiply  nine  hundredths  by  three  ten-thousandths  and 
divide  the   product  by  three  hundred   seventy-five  hundred- 
thousandths.      Then  multiply  the  quotient  by  six  hundredths 
and  subtract  the  result  from  seven  thousand  eight  hundred  fifty. 

3.  What  is  the  1.  c-  m.  of  the  three  prime  numbers  next 
after  19  ? 

4.  How  is  the  least  common  multiple  of  several-  prime 
numbers  found? 

5.  The  1.  c.  m.  of  6,  8,  and  12  is  what  per  cent  of  the  fourth 
multiple  of  12£? 

6.  What  is  the  largest  integer  that  will  exactly  divide  8,  12, 
and  16? 

7.  The  g.  c.  d.  of  18  and  24  is  what  per  cent  of  the  third 
multiple  of  8£? 

8.  Find  four  numbers  between  1  and  101,  each  of  which 
can  be  resolved  into  two  factors,  both  of  which  are  perfect 
squares  whose  roots  are  greater  than  1. 

9.  The  divisor  of  a  certain  number  is  189,  the  quotient  is  22, 
and  the  remainder  is  14|.     What  is  the  number  ? 

10.   Find  the  interest  of  $17.35  for  2  mo.  3  da.,  at  \°/o  per 
month. 


334       INVOLUTION  AND  EVOLUTION 

11.  Mr.  Rowe  bought  a  bill  of  goods  amounting  to  $500, 
receiving  a  discount  of  40  %  and  60  %  -     How  much  did  he  pay  ? 

12.  Mr.  E.   sold  $150  worth  of  goods  to  Mr.  W.,  taking 
Mr.  W.'s  note  for  the  amount  on  3  mo.  time.     The  note  was 
discounted  at  8%.     How  much  did  Mr.  K.  receive? 

13.  Mr.  A.  bought  120  U.   S.  3's  at   102,  brokerage  \%. 
These  bonds  are  non-taxable.     How  much  more  or  less  would 
he  gain  from  them  each  year,  than  from  the  same  amount  of 
money  loaned  at  6%  in  a  locality  where  the  tax  rate  is  37  mills 
on  a  dollar? 

14.  Mr.  N.  bought  1000  shares  of  Gas  and  Electric  Light  Co. 
stock  at  105.     He  kept  the  shares  until  a  semi-annual  dividend 
of  7J%  and  another  of  9%  had  been  paid,  and  sold  them  at  110. 
How  much  did  he  gain? 

15.  Mr.  Walker  bought  a  horse  for  $  75  and  sold  it  at  a  gain 
of  33^%.     He  took  a  note  for  the  amount,  due  in  one  year,  and 
had   the   note  discounted  at  10%.     What  per  cent  did  Mr. 
Walker  realize  ? 

16.  A  boy  had  80%  of  a  dollar,  spent  80%  of  what  he  had, 
and  lost  50%  of  what  remained.      What  per  cent  of  the  dollar 
did  he  then  have  ? 

17.  A  lawyer  having  a  debt  of  $1346.50  to  collect,  compro- 
mised by  taking  80  % .     His  fee  was  5  %  of  the  amount  collected. 
What  was  his  fee  and  how  much  should  he  return  to  his  client  ? 

18.  A  farmer  sold  25%  of  a  tract  of  land  containing  120  A., 
at  $.50  a  square  rod.     How  much  did  he  receive  for  it? 

19.  The  largest  bell  in  the  world  is  in  Moscow.     It  weighs 
216  T.     If  77%  of  it  is  copper,  and  the  rest  tin,  how  many  tons 
of  each  are  in  the  bell  ? 

20.  A  bell  in  Burmah  weighs  117  T.     If  it  contains  the 
same  proportion  of  tin  as  the  Moscow  bell,  how  much  tin  is 
there  in  it  ? 

21.  A   bell   in  Pekin  containing  the  same  proportions  of 
copper  and  tin  weighs  53  T.     How  much  copper  is  there  in  it  ? 


MISCELLANEOUS  EXEfcCISES  335 

22.  25  bu.  of  lime  were  bought  for  $6.25.      At  what  price 
per  peck  must  it  be  sold  to  gain  66f  %  ? 

23.  In  a  war  the  Brazilians  lost  43,365  men,  which  was  35% 
of  their  army.     How  many  were  left  in  the  army  ? 

24.  Mr.  C.'s  agent  in  N.  Y.  bought  for  him  a  bill  of  goods 
amounting  to  $  2575,  charging  him  a  commission  of  3%%.   How 
much  must  Mr.  0.  remit  to  pay  all  expenses? 

25.  An  agent's  commission  at  3%   upon  a  sale  was  $99. 
For  how  much  was  the  property  sold?     How  much  did  the 
owner  receive? 

26.  How  much  must  an  importer  pay  as  duty  on  7  casks  of 
wine,  each  containing  42  gal.,  2%  being  allowed  for  leakage, 
and  there  being  a  specific  duty  of  $1.25  per  gallon? 

27.  3  gal.  1  qt.  2  gi.  is  16f  %  of  how  much? 

28.  A  sold  goods  for  $70,  making  a  profit  of  16f  %.     What 
per  cent  would  he  have  made  by  selling  them  for  $  72  ? 

29.  After  retaining  3%  for  selling  my  potatoes,  my  agent 
sends  me  $523.80.     For  how  much  did  he  sell  them? 

30.  The  smaller  of  two  numbers  is  359.7  and  the  difference 
is  28 J.     What  is  the  larger  number? 

31.  Multiply  the  square  of  ^  by  the  reciprocal  of  -J-. 

32.  Multiply  f  by  the  square  of  the  reciprocal  of  f. 

33.  Multiply  (|)2  by  the  cube  of  the  reciprocal  of  j. 

34.  How  many  times  is  the  square  of  5  contained  in  the 
square  of  10  ? 

35.  How  many  times  is  the  square  of  6  contained  in  the 
square  of  12  ? 

36.  How  many  times  is  the  square  of  any  number  contained 
in  the  square  of  twice  that  number  ?     Illustrate. 

37.  How  many  times  is  the  cube  of  3  contained  in  the  cube 
of  6  ?     The  cube  of  5  in  the  cube  of  10  ? 

38.  How  many  times  is  the  cube  of  any  number  contained  in 
the  cube  of  twice  that  number  ?     Illustrate. 


336  INVOLUTION  AND   EVOLUTION 

39.  How  many  times  does  a  cube  of  a  number  contain  the 
cube  of  half  that  number  ?     Illustrate. 

40.  Make  a  true  equation  about  the  number  10  with  two 
terms  in  each  member.     Let  one  of  the  coefficients  be  7. 

Solve: 

41.  8z-5o;-4  =  31.  42.    9#  -  5z  +  30  =  58. 

43.  3x  +  Sx  +  6x-5  +  2x  —  8  =  31-x. 

44.  What  number  increased  by  5  times  itself  equals  42  ? 

45.  Each  of  the  sides  of  an  isosceles  triangle  is  -f  as  long  as 
the  base.     The  perimeter  is  34  in.     How  long  is  each  side  ? 

46.  The  perimeter  of  a  trapezium  is  54  in.     The  first  side  is 
J  as  long  as  the  second,  the  third  is  1J  times  the  second,  the 
fourth  is  1 J  times  the  second.     Find  the  length  of  each  side. 

47.  What  principal  will  gain  $  108  in  3  yr.  at  5%  ? 

48.  In  what  time  will  $  800  gain  $  108  at  6%  ? 

49.  At  what  rate  will  $  900  gain  $  78  in  2  yr.  9  mo. 

50.  What  principal  will  amount  to  $  629.20  in  3  yr.  6  mo. 
at  6%  ? 

51.  What  is  the  present  worth  of  $635.60  due  in  2  yr.  3 
mo.,  when  money  is  worth  6%  ? 

52.  What  number  is  that  to  which  if  its  ^  and  its  £  be  added 
the  sum  is  33  ? 

53.  From  a  rectangular  field  20  rd.  long  and  16  rd.  wide,  a 
man  bought  a  square  lot  which  was  ^  of  the  field.     What  was 
the  cost  of  fencing  the  lot  at  $  4.95  per  rod  ? 

54.  In  the  Webster  School  building  there  is  a  flight  of  stairs 
in  which  each  step  is  6  in.  high  and  10  in  wide,  each  step  pro- 
jecting 2  in.  over  the  edge.     There  are  13  steps  in  the  flight. 
Find  the  length  of  the  handrail. 

55.  How  much  would  it  cost  to  carpet  the  same  flight  of 
stairs  with  carpet  at  $  .90  per  yard,  allowing  4  in.  for  turning 
under  at  top,  the  same  at  the  bottom,  and  3  in.  for  each  turn  at 
the  edge  of  the  steps  ? 


MISCELLANEOUS  EXERCISES 


337 


56.   How  long  is  the  perimeter  of  a  right  triangle  whose 
base  is  60  in.  and  altitude  61  in.  ? 


A        F 

E 

N 


H 


M  L 

FIG.  9. 


57.  ABCD  is  a  12-inch  square.     AF, 
BH,  CL,  and  DN  are  each  4  in.     EA, 
OB,  KG,  and  MD  are  each  3  in.     How 
long  is  the  perimeter  of  the  irregular 
octagon  EFGHKLMN?    What  is   its 
area? 

58.  The  diagonal  of  a  square  is  12  in. 
Find  its  perimeter. 

59.  The  diagonal  of  a  rectangle  is  51  in.,  and  the  width  of 
the  rectangle  is  24  in.     Find  the  area  and  the  perimeter  of  the 
rectangle. 

60.  How  long  is   a  quadrant  of    a  circumference  whose 
diameter  is  21  in.  ? 

61.  ABCD  is  a  10-inch  square.    With  the 
vertex  of  each  angle  as  a  center  and  with 
a  radius  of  31  in.  an  arc  is  drawn.     How 
long  is  the  perimeter  of  each  sector  formed 
by  the  arc  and  the  parts  of  the  sides  of  the 
square  ?     How  long  is  the  perimeter  of  the 
irregular  figure  left  after  the  sectors  are 
cut  from  the  square  ? 

62.  A  chord  is  drawn  across  a  circle  in 
such  a  way  as  to  divide  the  circumference  into  two  arcs,  one  of 
which  is  four  times  the  other.     If  the  circumference  is  85  cm., 
how  long  is  each  arc  ? 

When  a  body  falls  from  an  elevated  place,  if  it  is  not  hindered  by  the 
air  or  other  obstructions,  it  falls  16T^  ft.  in  the  first  second,  and  3  times  as 
far  in  the  second  second.  In  the  third  second  of  time  it  falls  5  times  as  far 
as  in  the  first  second.  In  the  fourth  second  it  falls  7  times  as  far  as  in  the 
first  second,  and  so  on,  the  ratio  of  the  distance  passed  over  in  the  first 
second  to  the  distance  passed  over  in  any  given  second  being  equal  to  the 
ratio  of  1  to  the  corresponding  odd  number. 

HORN.    GRAM.    SCH.    AR.  —  22 


FIG.  10. 


338  INVOLUTION   AND   EVOLUTION 

63.  Develop  the  following  table  to  the  fifth  second  of  time : 
Distance  passed  over  in  1st   sec.  equals  16^  ft. 
Distance  passed  over  in  2d  sec.  equals  3  times  16TL  ft. 
Distance  passed  over  in  3d  sec.  equals  5  times  IGyL  ft. 

64.  Through  what  distance  will  a  falling  body  pass  in  the 
fourth  second  of  its  fall?     In  the  fifth  ? 

65.  How  far  does  such  a  body  fall  in  the  first  three  seconds? 
In  the  next  three  seconds  ? 

66.  1024  sq.  ft.  can  be  arranged  either  as  a  perfect  square  or 
as  a  rectangle  64  ft.  long,  or  128  ft.  long,  or  256  ft.  long.    What 
would  be  the  length  of  the  perimeter  in  each  case  ? 

67.  1296  sq.  ft.  can  be  arranged  either  as  a  perfect  square  or 
as  a  rectangle  72  ft.  long,  or  216  ft.  long,  or  432  ft.  long,  or 
648  ft.  long.     Find  the  length  of  the  perimeter  in  each  case. 

It  will  be  seen  that  when  a  given  area  is  arranged  in  the  form  of  a 
square,  it  has  a  shorter  perimeter  than  when  it  is  arranged  in  the  shape 
of  any  other  rectangle,  and  that  as  the  ratio  of  the  length  to  the  width 
increases,  the  perimeter  of  the  figure  also  increases.  Illustrate  this  fact. 

68.  A  farmer  has  a  rectangular  field  160  rd.  long  and  40  rd 
wide,  which  he  wishes  to  fence  and  divide  by  fencing  into  4 
equal  lots.     If  the  fencing  costs  $  .60  a  rod,  what  will  be  the 
difference  between  the  expense  of  running  his  dividing  fences 
parallel  with  the  long  sides  of  the  field,  and  the  expense  of 
running  them  parallel  with  the  short  sides  of  the  field? 

69.  Mrs.  Wood  has  a  rectangular  hall  35  ft.  long,  and  14  ft. 
wide,   the    floor    of    which    is   laid   with  parquetry  flooring. 
The  border  is  3£  ft.  wide.      At  $  1.25  per  square  yard,  what  is 
the  cost  of  the  floor  surface  inside  the  border? 

70.  The  floor  of  Mrs.  Wood's  sitting  room,  which  is  square, 
is  covered  with  the  same  kind  of  flooring,  and  has  a  border  of 
the  same  width.      The  surface  inside  the  border  in  the  sitting 
room  is  equal  in  area  to  the  surface  inside  the  border  of  the 
hall  floor.     If  the  cost  of  the  border  is  $2.75  per  linear  yard, 


MISCELLANEOUS   EXERCISES  339 

measured  around  the  edge  of  the  room,  how  much  more  will  the 
border  in  the  hall  cost  than  that  in  the  sitting  room? 

71.  Find  the   area  of  the  largest  rectangle   that  can  be 
inclosed  by  a  line  48  in.  long. 

72.  In  a  box,  each  of  whose  inside  measurements  is  9  in., 
there  were  packed  blocks  enough  to  make  3  cubes.      One  was 
a  6  in.  cube,  another  was  an  inch  cube.     How  long  was  the 
edge  of  the  third  cube  ? 

73.  How   many  cubic   inches   are  there   in  a   cube  whose 
edge  is  i^  of  a  foot? 

74.  A  pan  in  the  shape  of  a  rectangular  prism,  11  in.  long, 
and  7  in.  wide,  was  out  doors  during  a  rain  storm.     After  the 
storm  the  pan  was  found  to  contain  a  gallon  of  water.      How 
deep  was  the  water  in  the  pan? 

231  cu.  in.  =  1  gal. 

75.  Extract  the  square  root  of  2704.     Of  60516.     Of  1^_. 
Of  £ff     Of  44.89. 

76.  How  much  is  x  when  3  y?  =  75  ?     432?     243  ? 

77.  If  a  rectangular  lot  is  twice  as  long  as  it  is  wide,  and  its 
width  is  represented  by  x,  how  is  its  area  represented  ?     If  the 
area  is   72   sq.    in.,   what   are   the    dimensions    of    the    lot? 
Represent. 

78.  Figure  11  is  composed  of 
three  equal  squares  so  placed  that 
BC=  BF  and  DE  =  DH.  If  the 
area  of  the  whole  figure  is  147  sq. 
in.,  how  long  is  its  perimeter? 
'  79.  A  rectangular  lot  is  5 
times  as  long  as  it  is  wide,  and  it 
contains  80  sq.  rd.  .  How  many 

trees  can  be  placed  on  its  edge,  the  distance  from  center  to 

center  of  each  tree  being  1  rod?     Represent. 

80.   How  long  is  the  radius  of  the  largest  circle  that  can  be 
drawn  on  a  piece  of  paper  8  in.  long  and  6  in.  wide  ? 


CHAPTER  XI 

PROPORTION 

1.  What  number  has  the  same  ratio  to  6  that  1  has  to  3? 
That  2  has  to  3  ?     That  4  has  to  3  ? 

2.  Supply  the  missing  terms  : 

3  _  x          5  _  x          x  _  5  7    _  x 

5~10         9~27         4~10         15~90* 

3.  Which  of  the  following  statements  are  untrue  ? 

3:6  =  5:10.         8:4  =  12:6.         9:3  =  7:5.         1:3  =  3:15. 

4.  An  equation  which  states  the  equality  of  two  ratios  is 
called  a  Proportion. 

Write  a  proportion. 

5.  Write  a  proportion  with  the  terms  4,  6,  8  and  12,  and 

show  that  it  is  true. 

6.  Can  you  write  more  than  one  true  proportion  with  the 
terms  4,  6,  8,  and  12  ? 

7.  Write  a  proportion  with  the  terms  15,  20,  10,  and  30. 
With  4,  24,  7,  and  42.     With  7,  9,  21,  and  27. 

8.  Substitute  values  for  x  and  y  that  will  make  true  pro- 
portions. 

10:5  =  *:y.        8  : 12  =  x:y.        —=-•        80  :  40  =  x  :  y. 

Zi      y 

9.  The  terms  of  each  ratio  form  a  Couplet  of  which  the  first 
term  is  called  the  Antecedent  and  the  second  term  the  Consequent. 

In  the  proportion  8  : 16  =  9  : 18,  which  are  greater,  the  ante- 
cedents or  consequents  ? 

340 


PROPORTION 

10.  Write  a  proportion  in  which  the  antecedent  of  the  first' 
couplet  is  12  and  the  antecedent  of  the  second  couplet  is  8. 

11.  Write  a  proportion  in  which  the  consequent  of  the  first 
couplet  is  10  and  the  consequent  of  the  second  couplet  is  30. 

12.  Can  you  write  a  true  proportion  in  which  the  antecedent 
of  the  first  couplet  is  greater  than  its  consequent  and  the 
antecedent  of  the  second  couplet  is  less  than  its  consequent  ? 
Illustrate  and  explain. 

13.  Of  how  many  terms  must  a  proportion  consist  ? 

14.  The  first  and  fourth,  or  outside  terms  of  a  proportion, 
are  called  the  Extremes.     The  second  and  third,  or  inner  terms, 
are  called  the  Means. 

In  the  proportion  10  :  5  =  14  :  7  find  the  product  of  the 
means  and  the  product  of  the  extremes  and  compare  them. 

15.  Write  some  proportions  and  compare  the  product  of  the 
means  with  the  product  of  the  extremes  until  you  see  the  truth 
of  the  following  principle  : 

In  a  proportion  the  product  of  the  means  equals  the  product  of 
the  extremes. 

When  three  terms  of  a  proportion  are  given  the  other  term  is  easily 
found  by  this  principle,  as  : 

3  :  6  =  25  :  x.  The  product  of  the  means  is  3  x.  The  product  of  the 
extremes  is  6  x  25.  As  these  products  are  equal  we  have  the  equation 
3«  =  6  x25. 


Find  the  missing  term  in  each  of  the  following  proportions  : 

16.  27:40  =  9:#.  19.    144:^  =  12:1. 

17.  24  :  3  =  48  :  x.  20.    9  :  6  =  x  :  12. 

18.  25:z  =  35:7.  21.    11:77  =  ^:42. 

A  double  colon  is  sometimes  used  between  the  ratios  instead  of  the  sign 
of  equality,  as  6  :  4  :  :  3  :  2.     This  is  read  6  is  to  4  as  3  is  to  2. 


342  PROPORTION 

22.  Find  the  value  of  x  in  the  proportion  x :  9  : :  10  : 18. 
Find  the  values  of  x : 

23.  x  :  36  : :  3  : 12.  30.  2| :  10  :  :  8J  :  x. 

24.  a  :  40::  18:  80.  31.  6£:  25  :  :  12|-  :  a. 

25.  21:aj::26:8J.  32.  6J  :  12}  :  :  76  :  a. 

26.  28:  35::  16:  a.  33.  16| :  33J  : :  66| :  x. 

27.  6$:  33$::  12:  a.  34.  6}  :  a  :  :  6  : 12. 

28.  7}:  22}::  5:  a.  35.  3J  :  50  :  :  a  :  9. 

29.  3J:7}::3:a;.  36.  16f  :  12}  : :  a  :  4. 

37.  What  is  the  ratio  between  30  min.  and  10  hr.  ?     Be- 
tween 3  Ib.  and  1  Ib.  8  oz.  ? 

Observe  that  a  ratio  is  only  possible  between  quantities  of  the  same 
denomination. 

Substitute  numbers  for  x  and  y  that  will  make  the  following 
proportions  true: 

38.  x  :  2  : :  12  :  y.  40.    x  :  6  :  :  3  :  y. 

39.  x:8::8:y.  41.    x  :  1 :  :  1 :  y. 

42.  A  proportion  in  which  the  means  are  equal  is  called  a 
Mean  Proportion,  and  the  number  which  each  mean  represents 
is  called  a  Mean  Proportional  between  the  other  two  numbers. 

Write  a  proportion  in  which  each  of  the  means  is  6. 

43.  In  the  mean  proportion  1 :  4  =  4  : 16,  what  number  is 
the  mean  proportional  ?     In  the  proportion  2  :  6  =  6  : 18,  6  is 
a  mean  proportional  between  what  numbers  ? 

Find  value  of  x : 

44.  3  :#::#:  12.  46.    2  :  x  :  :  x  :  8.  48.    7  :  x  :  :  x:  28. 

45.  !:»::«:  121.         47.    2  :  x  : :  x  :  98.        49.    3  :  x  :  :  x  :  27. 

50.    Write  several  proportions  in  which  each  of  the  means 
is  12. 


PROPORTION  343 

51.  If  3  hats  cost  $  11,  how  much  will  6  hats  cost? 

Let  x  =  the  number  of  dollars  paid  for  6  hats.  The  greater  the  number 
of  hats  the  greater  the  number  of  dollars  paid  for  them.  3  hats  are  to  6 
hats  as  $  1 1  (the  price  of  3  hats)  are  to  x  dollars  (the  price  of  6  hats)  or 
3:6  :  :  11  :X. 

52.  If  5  hats  cost  $  7,  how  much  will  10  hats  cost  ? 

In  each  problem  it  is  assumed  that  the  articles  considered  are  of  the 
same  kind  and  of  equal  value. 

53.  If  8  apples  cost  15^,  how  much  will  16  apples  cost  ? 

54.  If  9  apples  cost  17^,  how  much  will  3  apples  cost? 

55.  If  10  oranges  cost  53^,  how  much  will  5  oranges  cost? 

56.  If  7  hats  cost  $  25,  how  much  will  21  hats  cost  ? 

57.  If  3  vases  cost  $  26,  how  much  will  5  vases  cost  ? 

58.  If  9  pairs  of  opera  glasses  cost  $36,  how  much  will  11 
pairs  cost  ? 

59.  If  51  yd.  of  lace  cost  $  17,  how  much  will  16£  yd.  cost  ? 

60.  If  10  yd.  of  silk  cost  $  23,  how  much  will  13^  yd.  cost  ? 

61.  If  80  yd.  of  cloth  cost  $375,  how  much  will  5f  yd.  cost  ? 

62.  If  20  yd.  of  calico  cost  $  1.35,  how  much  will  6f  yd.  cost? 

63.  If  15  yd.  of  jet  cost  $7.50,  how  much  will  8£  yd.  cost? 

64.  In  the  proportion  5  : 10  :  :  12  :  24,  if  both  terms  of  the 
first  couplet  are  divided  by  5,  is  the  proportion  still  true  ? 
By  what   number  may  each  term   of  the  second  couplet  be 
divided  without  destroying  the  proportion? 

65.  If  9  men  can  earn  $  23  in  a  day,  how  much  can  18  men 
earn  ? 

66.  If  15  men  earn  $  37  in  a  day,  how  much  will  25  men 
earn? 

67.  If  30  men  earn  $  175  in  a  day,  how  much  will  20  men 
earn  ? 


344  PROPORTION 

68.  When  $19  are   paid  for  8   hats,  how  much  will   24 
hats  cost  ? 

69.  If  a  train  runs  85  mi.  in  3  hr.,  how  far  will  it  run  in 
21  hr.? 

All  problems  in  proportion  can  be  solved  by  analysis,  as  "If  a  train 
runs  85  mi.  in  3  hr.,  in  21  hr.  it  will  run  -^1-  times  85  mi.,  or  595  mi." 

Solve  the  following  problems  by  analysis  as  well  as  by  pro- 
portion : 

70.  John  rode  on  his  bicycle  5  mi.  in  45  min.     At  the  same 
rate,  how  far  would  he  ride  in  90  min.? 

Find  the  cost  of  10  articles  of  the  same  kind : 

71.  When  3  fans  cost  $  3.50.         74.   When  4  bags  cost  $  .75. 

72.  When  6  maps  cost  $.25.         75.    When  6  caps  cost  $  1. 

73.  When  8  tops  cost  $  .25.          76.    When  4  books  cost  $  5. 

77.  Mr.  A.  collected  $  125  in  the  first  four  days  of  the  week. 
During  the  rest  of  the  week  he  collected  at  the   same   rate. 
How  much  did  he  collect  in  the  whole  week  ? 

78.  From  80  A.  of  land  Mr.  Porter  gathered  1680  bu.  of 
corn.     At  the  same  rate,  how  many  bushels  could  he  gather 
from  167  A.  of  corn  land  ? 

79.  Polygons  which  have  the  same  shape  are  said  to  be 
Similar.     Their  corresponding  angles  are  equal,  and  their  cor- 
responding sides  are  proportional. 

There  are  two  similar  rectangles.  The  larger  rectangle  is 
18  in.  long  and  12  in.  wide.  The  smaller  rectangle  is  9  in. 
long.  How  wide  is  it  ?  Represent. 

80.  Draw  a  rectangle  8  in.  long  and  2  in.  wide.     Draw  a 
similar  rectangle  4  in.  long.     Find  the  ratio  of  the  perimeters 
of  the  rectangles. 


PROPORTION 


345 


12 


5  B       D 
FIG.  1. 


10 


81.  The  right  triangles  ABC  and  DEF 
are   similar.     Find  the  side  BC.     Then 
find  by  proportion  the  sides  DF  and  FE 
of  the  larger  triangle.     Find  the  ratio  of 
the  perimeters  of  the  triangles. 

82.  Find   the  hypotenuse   of   a   right 
triangle  whose  shorter  sides  are  7  in.  and 
24  in.     Find  the  length  of  each  side  of  a 

E   similar  triangle  whose  shortest  side  is  21 
in.     Find  the  ratio  of  the  perimeters. 

83.  The  perimeter  of  an  isosceles  triangle,  whose  base  is  5 
in.,  is  19  in.  How  long  is  each  of  the  equal  sides  ?  Find  the 
length  of  each  side  of  a  similar  triangle  whose  base  is  20  in. 
Represent.  Find  ratio  of  perimeters. 

84.  Find  the  length  of  each  side  of  a 
trapezoid  similar  to  ABCD,  but  larger, 
each  of  the  non-parallel  sides  of  the  larger 
trapezoid  being  8  in.  long.  What  is  the 
ratio  of  the  perimeter  of  the  greater  trape- 
zoid to  the  less  ? 


4  in. 


6  in. 
FIG.  2. 


Observe  that  the  ratio  of  the  perimeters  of  two  similar  polygons  is  the 
same  as  that  of  anj^  pair  of  their  corresponding  sides. 

85.  There  are  two  similar  trapeziums.  The  perimeter  of 
the  smaller  is  18  in.,  and  its  shortest  side  is  3  in.  Find  the 
perimeter  of  the  greater  trapezium,  its  shortest  side  being  15 

in.     Represent. 

86.    The  rectangle  AB CD  is 

twice  as  long  as  the  similar 
rectangle  EFGH,  and  all  the 
corresponding  lines  of  the  two 
rectangles  are  proportional. 
If  AB  is  24  in.,  EF 12  in.,  and 

DB  26  in.,  how  long  is  HF  ?    If  BC  is  10  in.,  how  long  is  FG  ? 

Find  the  area  of  each  rectangle  and  the  ratio  of  their  areas. 


346 


PROPORTION 


87.  There  are  two  similar  rectangles,  whose  perimeters  are 
respectively  36  in.  and  9  in.     If  the  base  of  the  larger  rec- 
tangle is  11  in.,  how  long  is  the  base  of  the  smaller  rectangle  ? 
If  the  altitude  of  the  larger  rectangle  is   7  in.,  what   is  the 
area  of  the  smaller  rectangle  ? 

88.  The  upper  base  of  a  certain  trapezoid  is  5  in.     Its  per- 
imeter is  17  in.     How  long  is   the   upper   base  of   a  similar 
trapezoid  whose  perimeter  is  51  in.?     If  the  altitude  of  the 
smaller  trapezoid  is  4  in.,  what  is  the  altitude  of  the  larger 
trapezoid  ? 

89.  There  are  two  similar  rhomboids,  a  base  of  one  being  12 
in.,  and  a  corresponding  side  of  the  other  6  in.     If  the  alti- 
tude of  the  larger  rhomboid  is  4  in.,  what  is  the  altitude  of  the 
smaller  ?     Eepresent. 


90.  The    pentagon  FQHJK   is    similar   to  the    pentagon 
ABODE.     Find  the  length  of  the  sides  which  are  unmarked. 

91.  The  perimeter  of  an  irregular  hexagon  is  330  in.     Its 
longest  side  is  72  in.     Find  the  length  of  the  longest  side  of  a 
similar  irregular  hexagon,  whose  perimeter  is  55  in. 

92.  If  6  men  can  do  a  piece  of  work  in  15  da.,  how  long  will 
it  take  2  men  to  do  the  work  ? 


PROPORTION  347 

In  cases  like  the  foregoing  the  proportion  is  inverse.  The  less  the 
number  of  men  employed,  the  greater  the  number  of  days  required. 
6  men  :  2  men,  not  as  15  da.  :  x  da.,  but  as  x  da. :  15  da.  We  have,  there- 
fore, the  proportion 

men     men  da.        da. 

6     :     2     :      :     x     :     15 

We  know  that  in  this  proportion  x  stands  for  a  number  greater  than  15 
because  6,  the  antecedent  of  the  first  couplet,  is  greater  than  2,  the  conse- 
quent of  that  couplet,  and  the  ratio  of  x  to  15  is  the  same. 

93.  If  30  men  can  do  a  piece  of  work  in  40  da.,  how  many 
men  are  required  to  do  the  work  in  20  da.  ?     In  10  da.  ?     In  5 
da.  ?     In  8  da.  ?    In  4  da.  ? 

In  solving  a  problem  by  proportion  it  is  necessary  first  to  determine 
whether  the  proportion  is  inverse  or  direct.  In  the  following  problems 
write  the  word  "  Direct "  or  "  Inverse  "  at  the  beginning  of  each  solution. 

94.  A  pole  40  ft.  high  casts  a  shadow  8  ft.  long.    How  long 
is  the  shadow  of  a  10-ft.  pole  at  the  same  time  and  pface  ?    Of 
a  20-f  t.  pole  ?     Of  a  16-f  t.  pole  ? 

95.  Twelve  days  are  required  for  a  piece  of  work,  if  the  men 
work  10  hr.  a  day.     How  many  days  will  be  required  if  they 
work  8  hr.  a  day  ?     9  hr.  a  day  ?     6  hr.  a  day  ? 

96.  The  taxes  upon  a  piece  of  property  valued  at  $  7800  are 
$  195.     At  the  same  rate,  what  is  the  amount  of  taxes  upon 
property  valued  at  $  5600  ? 

97.  When  a  factory  runs  18  hr.  a  day,  a  piece  of  work  is 
finished  in  20  da.     How  many  hours  a  day  must  the  factory 
run  to  finish  the  work  in  15  da.x?     In  24  da.  ?     In  30  da.  ? 

.     98.    Goods  costing  $  700  are  sold  for  $  800.     At  the  same 
rate  what  is  the  selling  price  of  goods  costing  $  1050  ?     $  1750  ? 
99.    If  63  gal.  of  vinegar  cost  $  12.60,  how  much  will  7  gal. 
cost  ?     9  gal.  ?    23  gal.  ?    10  gal.  ? 

100.  From  a  debt  of  $8000,  Mr.  A.  collects  $  7000.  At  the 
same  rate  how  much  will  he  collect  from  a  debt  of  $  4800  ? 
Of  $1600?  Of  $560?  Of  $60? 


348  PROPORTION 

101.  If  a  5^  loaf  of  bread  weighs  12  oz.  when  flour  is  $  6  per 
barrel,  how  much  should  it  weigh  when  flour  is  $  4  per  barrel  ? 
$  8  per  barrel  ?     $  3  per  barrel  ? 

102.  If  it  costs  $  60  to  cover  a  floor  with  carpet  costing  $  1.25 
per  yard,  how  much  would  it  cost  to  cover  it  with  carpet  at 
$  2  per  yard  ?.    $  1.75  per  yard  ?     $  1  per  yard  ? 

103.  The   carpet   which   covers   a  room   20  ft.  long   costs 
$  88.60.     What  would  be  the  cost  of  covering  with  carpet  of 
the  same  quality  a  room  of  the  same  width  but  5  ft.  longer  ? 
4  ft.  shorter  ? 

104.  A  stock  company  paid  a  semiannual  dividend  of  5%. 
Mr.  B.'s  dividends  were  $  800.     How  much  would  he  have 
received  if  the  rate  had  been  1%  ?     9%  ?     10% 


105.  TJie  amount  of  duty  upon  a  certain  importation  was 
$  216.64.     The  rate  was  40  %  ad  valorem.     If  the  rate  had  been 
55  °/o  ad  valorem  what  amount  of  duty  would  have  been  paid  ? 

106.  Mr.  L.'s  yearly  income  from  bonds  paying  3%  is  $  1275. 
What  would  be  his  yearly  income  from  those  bonds  if  they 
were  5%  bonds? 

107.  Mr.  K.  has  a  sum  of  money  invested  at  6%.     12  years' 
interest  on  that  sum  paid  for  his  house  and  lot.     How  many 
years'  interest  would  have  been  sufficient  to  pay  for  the  house 
and  lot  if  the  rate  of  interest  had  been  8%  ? 

PROPORTIONAL  PARTS 

108.  John  and  Harry  hired  a  boat  to  go  fishing.     John  paid 
30^.     Harry  paid  -f  as  much  as  John.     How  much  did  both 
pay  ?    What  part  of  the  whole  expense  did  each  pay  ?     They 
sold  their  fish  for  $  1.     How  ought  the  money  to  be  divided  ? 

109.  If  20^  are  divided  between  two  boys  so  that  the  first 
boy's  share  is  to  the  second  boy's  share  as  2  is  to  3,  how  much 
will  each  boy  receive  ? 


PROPORTIONAL  PARTS  349 

If  20  were  divided  into  5  equal  parts,  ought  not  the  first  boy  to  have  2 
of  these  parts,  or  f  of  the  whole,  and  the  second  boy  3  of  the  parts  or  f  of 
the  whole  ? 

110.  Separate  27  into  2  numbers  in  the  ratio  of  4  to  5. 

Combining  4  and  5  we  have  9  equal  parts  to  which  27  is  equal.  Each 
number  equals  how  many  9ths  of  27  ? 

111.  Divide  12  in  the  ratio  of  5  to  1.     In  the  same  way  divide 
36.     54.     84. 

112.  How  would  you  divide  14^  between  2  boys,  giving 
one  boy  6  times  as  much  as  the  other  ? 

113.  How  must  $9.9  be  divided  among  3  men  that  their 
shares  may  be  in  the  proportion  of  1,  3,  and  5  ? 

114.  Three  men  are  to  receive  $54  for  papering  a  house. 
The  first  man  has  worked  5  da.,  the  next  6  da.  and  the  next 
7  da.     How  much  shall  each  receive  ? 

115.  The  bill  for  the  labor  of  shingling  a  house  is  $48. 
How  shall  that  be  divided  among  the  three  workmen  A,  B,  and 
C,  if  A  has  worked  6  da.,  B  4  da.,  and  C  2  da.  ? 

116.  Two  painters  received  $  51  for  painting  a  house.     The 
first  worked  10  da.,  and  the  other  worked  7  da.     How  ought 
the  money  to  be  divided  ? 

117.  Mr.  A.,  Mr.  B.,  and  Mr.  C.  sent  some  poor  children  to 
spend  a  week  in  the  country.     Mr.  A.  paid  for  5  children,  Mr. 
B.  for  8  children,  and  Mr.   C.   for  11  children.     The  whole 
expense  was  $  54.72.     How  much  did  each  man  pay  ? 

118.  In  a  pasture  there  were  40  cows  of  which  Mr.  D. 
owned  7  cows,  Mr.  E.  21  cows,  and  Mr.  F.  the  remainder.     If 
the  cost  of  pasturage  was  $  70  a  month,  how  much  did  each 
owner  of  the  cows  pay  each  month  ? 

119.  Two  men  contract  to  do  a  piece  of  work  for  $  122.50. 
The  work  is  finished  in  20  days,  but  one  man  is  unable  to  work 
for  5  days.     How  shall  the  amount  paid  be  divided  ? 


350  PROPORTION 

120.  A  and  B  sent  in  a  bid  for  the  plumbing  of  a  house  for 
$  150,  but  afterwards  reduced  the  price  $  14.     The  work  was 
finished   in  16  da.   after   its  commencement.     They  hired   a 
helper  at  $  2.50  per  day  who  worked  every  day.     A  worked 
every  day  except  one,  and  B  was  absent  7  da.     How  shall  the 
amount  be  divided  ? 

121.  Gunpowder  is  made  of  75  parts  of  saltpeter,  15  of 
charcoal,  and  10  of  sulphur.     How  many  pounds  of  each  are 
there  in  1£  T.  of  gunpowder? 

122.  Two  young  men  hired  a  boat  for  $  1.25.     One  invited 
two  friends,  and  the  other  invited  one  friend  to  go  in  the  boat 
with  them.     How  should  the  young  men  divide  the  expense  ? 

123.  Divide  150  into  3  parts  in  the  ratio  of  3,  4,  and  8. 

124.  Mrs.  A.  had  three  children,  one  5  yr.,  one  7  yr.,  and 
one  13  yr.  old.     She  proposed  to  divide  a  bag  of  nuts  among 
the  children  in  the  ratio  of  their  ages,  if  the  eldest  would  tell 
correctly  how  they  should  be  divided.     In  that  case,  how  many 
nuts  would  each  child  receive,  if  there  were  75  nuts  in  the  bag  ? 

125.  Mr.  Allen  and  Mr.  Ward  formed  a  partnership,  Mr. 
Allen  putting  in  $  1200,  and  Mr.  Ward  $  2400.     They  gained 
$  2700  the  first  year.     How  ought  the  money  to  be  divided  ? 

126.  The  next  year  Mr.  Ward  put  in  $  600  more,  and  the 
profit  was  $  3000.     Find  each  man's  share. 

127.  The  next  year  Mr.  Allen  took  out  $  200,  and  Mr.  Ward 
put  in  $  1000.     The  profit  was  $  3500.     As  Mr.  Allen's  share 
of  the  profits  was  so  small  and  he  was  giving  much  time  to 
the  business  he  was  paid  $  500  from  the  profits.     How  much 
did  each  man  receive  ? 

.128.  Two  brothers  bought  a  $5000  United  States  bond  at 
105,  the  younger  brother  furnishing  J  as  much  of  the  money 
as  the  elder.  How  much  did  each  pay  ? 


MISCELLANEOUS  EXERCISES 


351 


129.  Mr.  A.  and  Mr.  B.  hired  a  pasture  for  $  76.     Mr.  A. 
put  in  8  cows  for  9  weeks,  and  Mr.  B.  10  cows  for  8  weeks. 
How  much  ought  each  man  to  pay  ? 

130.  CLASS  EXERCISE.        —  may  make  a~  problem  in  which 
a  number  of  dollars  are  unequally  divided  among  a  number  of 
persons,  and  the  class  may  solve  it. 

MISCELLANEOUS  EXERCISES 

1.  92:32  =  62:? 

2.  7V529  =  ?     4Vl521  =  ?     8V1369  -  6- 

3.  V2025  :  VT225  =  ? 

4.  V3025  :  V4225  ::!!:? 

V2304  :  ? 
3V784:? 


5.  V1764  :  V2401  : 

6.  4Vl96:7V576 

7.  83:16::48:? 

8.  V: 

9.  V729  :  3V2916  :  : 


1296  :  ? 

10.  How  many  cubic  inches  in  a  cube  whose  surface  contains 
96  sq.  in. 

11.  How  many  acres  of  land  in  a  road  10  mi.  long  and  55  ft. 
wide? 

Use  cancellation. 

12.  A  lake,  whose  area  is  45  A.  is  covered  with  ice  3  in. 
thick.     Find  the  weight  of  the  ice  in  tons,  if  a  cubic  foot  of 
ice  weighs  920  oz. 

13.  A  room  10  ft.  high  contains  30,000  cu.  ft.     How  much 
will  it  cost  to  carpet  it  at  75  ^  per  square  yard  ? 

14.  Sunset  Park  contains  115  A.  of  land.     How  much  is  it 
worth  at  $  .12|  a  square  foot  ? 

15.  A  man  bought  a  cow  and  paid  $  20.25  cash,  which  was 
90%  of  the  cost.     How  much  did  the  cow  cost? 


352  PROPORTION 

16.  Change  .15  and  .025  to  common  fractions  in  their  lowest 
terms. 

17.  Eeduce  to  lowest  terms  |f  J. 

18.  Eednce  5  to  a  fraction  having  11  for  a  denominator. 

19.  Change   .37    and    .0016   to   decimals   having    common 
denominators. 

7.5  x  .25  x  3.6  x  .18      =? 
'    .009  x  .08  x  5  x  .125  x  .3 

21.  Divide  the  product  of  13.5  and  1.8  by  8.1. 

22.  A  farmer  sold  8J  cords  of  wood  at  $  5  per  cord,  and 
received  an  equal  amount  of  money  from  selling  apples  at 
$  1.25  per  bushel.     How  many  bushels  did  he  sell  ? 

23.  Mr.  Brown  bought  47 £  T.  of  coal  at  $  6f  a  ton.     He 
paid  cash  $  175,  and  gave  his  note  for  the  balance  payable  in 
3  mo.  with  grace.     What  were  the  proceeds  of  the  note  dis- 
counted at  6%  ? 

24.  Make  out  and  receipt  a  bill  for  the  following :  May  1, 
1898,  James  Bentry  of  Ft.  Wayne,  Ind.,  bought  of  H.  A.  Cook 
&  Son,  16  Ib.  of  tea  at  $.85  per  pound,  36  Ib.  of  coffee  at  $  .18} 
per  pound,  8  packages  of  macaroni   at   $  .12J  per  package, 
3  gal.  strawberries  at  $  .25  per  gallon,  25  loaves  of  bread  at 
$  .05  per  loaf. 

25.  Find  the  l.c.m.  of  18,  24,  15,  30. 

26.  Find  the  prime  factors  of  342. 

27.  The  sum  of  two  numbers  is  219.5,  and  one  of  the  num- 
bers is  96.875.     What  is  the  other  number  ? 

28.  If  a  table  costs  as  much  as  two  chairs,  and  five  chairs 
cost  $  43.75,  how  much  does  the  table  cost  ? 

29.  Draw  a  representation  of  a  cord  of  wood,  marking  its 
dimensions. 

30.  What  per  cent  is  gained  by  buying  peanuts  at  $  2  a 
bushel  and  selling  them  at  5^  a  pint  ? 


MISCELLANEOUS   EXERCISES 

. 

* 


31.  How  mucli  will  it  cost  to  fence  a  rectangular  l 
long,  containing  425  sq.  rcL,  if  the  fence  costs  20^  a  foot  ? 

32.  The  dimensions  of  a  bin  are  1  ft.  3  in.  by  3  ft.  4  in.  by 
9  ft.  4  in.     How  many  bushels  of  wheat  will  it  hold  ? 

There  are  2150.4  cu.  in.  in  a  bushel. 

33.  At  15^  a  square  yard,  how  much  will  it  cost  to  paint 
the  walls  and  ceiling  of  a  room  36  ft.  long,  24  ft.  wide,  and 
12  ft.  high,  having  a  baseboard  9  in.  high.     No  allowance  for 
openings. 

34.  When  5  children  had  left  a  class,  75%  of  the  class  re- 
mained.    How  many  children  belonged  to  the  class  at  first  ? 

35.  If  envelopes  are  bought  at  the  rate  of  15  cents  for  a 
package  of  25  and  sold  for  a  cent  apiece,  what  per  cent  is 
gained  ? 

36.  From  a  box  containing  a  dozen  packages  of  envelopes, 
100  envelopes  were  used.     What  per  cent  of  the  number  was 
left? 

37.  After  a  battle,  the  number  of  soldiers  who  answered  to 
their  names  at  roll  call  was  only*  612,  which  was  60%  of  the 
number  that  went  into  battle.     How  many  went  into  battle  ? 

38.  There  were  72  bananas  in  a  bunch  that  cost  85^.    12-^-% 
of  them  were  sold  at  3^  apiece,  33^%  at  15^  a  dozen,  and 
S^%  of  them  were  spoiled.     The  rest  were  sold  at  2^  each. 
What  was  the  gain  on  the  whole  bunch  ? 


39.  From  a  school  of  48  pupils,  8  J%  were  absent  on  a  rainy 
day.     25%  of  those  present  went  out  of  the  room  to  a  recita- 
tion.    How  many  remained  in  the  room  ? 

40.  A  bar  of  fresh  soap  weighed  3  Ib.  6  oz.     When  dry,  it 
weighed  33^%  less.     How  much  did  it  weigh  then  ? 

41.  Walter  bought  a  knife  for  50^,  and  exchanged  it  for  a 
school  book  worth  67^.     What  per  cent  did  he  gain  ? 

HORN.    GRAM.    SCH.    AR.  -  23 


354  PROPORTION 

42.  A  jeweler  bought  some  pins  at  the  rate  of  $24.00  a 
dozen.     The  cost  of  each  was  66 f%  of  the  price  for  which 
he   sold   them.     What  was   the  selling  price   of  each   pin? 
What  per  cent  of  profit  did  he  make  ? 

43.  A  shoe  dealer  sold  a  pair  of  shoes  for  $  4.00,  gaming 
33^%.     How  much  did  they  cost  ? 

44.  On  the  4th  of  July  Alfred  had  half  a  dollar  and  James 
had  30  cents.     They  put  their  money  together,  and  spent  30% 
of  it  for  firecrackers,  25%  of  the  remainder  for  candy,  and  the 
rest  for  lemonade.     How  much  did  they  spend  for  each  ? 

45.  John's  arithmetic  when  new  cost  $  .60.     When  he  had 
finished  the  study  of  arithmetic,  he  sold  the  book  for  $  .40. 
What  was  the  per  cent  of  reduction  ? 

46.  Out  of  3  dozen  trees  that  a  gardener  set  out,  only  24 
trees  lived.     What  per  cent  died  ? 

47.  Cranberries  sold  at  15^  a  quart  brought  a  gain  of  20%. 
How  much  did  they  cost  per  bushel  ? 

48.  A  man  who  had  $  675  spent  21%  of  it  for  a  coat,  and 
put  66f  %   of  the  remainder  in  bank.      Find  price  of  coat, 
amount  of  money  in  the  bank,  and  money  left. 

49.  A  housekeeper  used  8^  Ib.  from  a  bag  of  flour  leaving 
66f  %  of  it.     How  many  pounds  were  left  ? 

50.  20%  of  Mr.  Ward's  farm  is  planted  in  corn  and  50% 
in  wheat.     The  rest,  which  is  36  acres,  is  pasture  land.     How 
many  acres  are  there  of  corn  ?     Of  wheat  ? 

51.  Mr.  A.  bought  a  horse  for  $  350  and  sold  it  to  Mr.  B.  at 
a  gain  of  10%.    Mr.  B.  sold  it  to  Mr.  C.  at  a  gain  of  5%.    How 
much  more  did  Mr.  C.  pay  for  the  horse  than  Mr.  A.  paid  ? 

52.  100  yd.  of  carpet  which  cost  $60  to  manufacture  were 
sold  by  the  manufacturer  to  the  wholesale  dealer  at  a  profit  of 

The  wholesale  dealer  sold  them  at  a  profit  of  25%. 


MISCELLANEOUS   EXERCISES  865 

The  retail  dealer  sold  them  to  a  customer  at  a  profit  of  25%. 
What  was  the  retail  price  per  yard  ?  How  much  more  per  yard 
did  the  customer  pay  than  the  manufacturer  received  ? 

53.  A  dealer  lost  16f%  by  selling  goods  for  $500.     What 
was  the  cost  ? 

54.  A  boy  sold  his  bicycle  for  $31.25,  gaining  5%.     How 
much  did  the  bicycle  cost  him  ? 

55.  By  selling  a  bicycle  for  $40  a  dealer  gained  33|%. 
How  much  would  he  have  gained  by  selling  it  for  $  50  ? 

56.  21  is  what  per  cent  of  1\  ?     25  ?     8J  ?     16|  ? 

57.  What  per  cent  is  gained  by  buying  berries  at  the  rate  of 
24  qt.  for  a  dollar  and  selling  them  at  the  rate  of  4  qt.  for 
25^? 

58.  What  per  cent  is  gained  by  buying  fans  at  the  rate  of  $  1 
a  dozen  and  selling  them  at  10  ^  apiece  ? 

59.  Mr.  X.  bought  a  stock  of  groceries  for  $  1200.     He  sold 
J  of  them  at  a  profit  of  33|%,  £  of  them  at  a  profit  of  25%, 
and  lost  10%  on  the  remainder.     Did  he  gain  or  lose  on  the 
whole,  and  how  much  ?     What  per  cent  ? 

60.  A  lawyer  collected  60%  of  a  debt  of  $2400,  receiving 
10%  for  collecting.     He  afterward  collected  75%  of  the  re- 
mainder, for  which  he  was  paid  at  the  same  rate.     How  much 
of  the  debt  was  paid  to  the  creditor,  how  much  did  the  lawyer 
receive,  and  how  much  did  the  debtor  fail  to  pay  ? 

61.  A  collector  who  received  5%  for  his  services  earned  in 
one  day  $  17.50.     How  much  did  he  collect  ? 

62.  A  store  valued  at  $  25,000  was  insured  for  £  of  its  value 
at  |  % .     What  was  the  premium  ? 

63.  The  premium  for  insuring  a  house  for  ^  of  its  value  at 
1  %  was  $  16.     What  was  the  value  of  the  house  ? 

64.  The  premium  for  insuring  a  building  for  £  of  its  value 
at  1£%  was  $  60.     What  was  the  value  of  the  building  ? 


356  PROPOKTION 

65.  Mr.  S.  has  real  estate  assessed  at  $  8000  in  a  city  where 
the  tax  rate  is  $  .033  on  the  dollar,  and  the  poll  tax  is  $  2. 
How  much  are  his  taxes  ? 

66.  The  taxes  of  Mr.  T.,  who  lives  in  the  same  city,  are 
$  233,  of  which  $  2  is  his  poll  tax.     For  how  much  is  his  prop- 
erty assessed? 

67.  What   is   the  duty  on  750  yd.  of  cloth  invoiced  at  5 
francs  a  yard,  a  franc  being  19T\^,  the  duty  being  30%  ad 
valorem  ? 

68.  An  importer  paid  $  7000  in  duties  upon  an  importation. 
The  duty  was  25%  ad  valorem.     How  many  dollars'  worth  of 
goods  did  he  import  ? 

69.  A  man  failing  in  business  paid  $  .40  on  the  dollar.     He 
owed   Gay   &   Co.   $8756,  W.   H.  Reed   $10,857,  the   First 
National  Bank  $  5000,  and  Hall  &  Co.  $  4221.    How  much  did 
each  receive  ? 

70.  A  man  failed  in  business  owing  $24,000,  and  having 
assets  $  12,000.  What  per  cent  of  his  debts  could  he  pay  ?  How 
much  would  a  creditor  receive  to  whom  he  owed  $  1785.50  ? 

71.  How  many  cents  on  a  dollar  can  each  of  the  bankrupts 
in  the  following  list  pay  ? 

Liabilities         Assets  Liabilities          Assets 

Mr.  Low,   $84,000     $46,200      Mr.  Dow,   $125,600     $37,680 
Mr.  Van,    $96,800     $33,880      Mr.  May,   $242,850     $97,140 

72.  Mr.  Wright  is  a  creditor  of  Mr.  Low  to  the  amount  of 
$7000.      Mr.  Van  owes  Mr.  Wright  $6400.     Mr.  Dow  owes 
him  $  475,  and  Mr.  May  owes  him  $  2821.     How  much  will  he 
receive  in  the  settlement  of  those  debts  ? 

73.  Mr.  L.  paid  $  800  for  a  lot,  and  f  as  much  for  another 
lot.     He  sold  the  higher  priced  lot  at  a  gain  of  62i%,  and  the 
cheaper  lot  at  a  loss  of  10%.     How  much  did  he  gain  on  the 
whole  investment  ?     What  per  cent  ? 


MISCELLANEOUS  EXERCISES  357 

74.  Goods  invoiced  at  $  837  were  discounted  30%  and  20%. 
What  was  the  net  price  ? 

75.  The  list  price  of  a  bill  of  hardware  was  $  90.     The  dis- 
counts were  60%  and  25%.     The  goods  were  sold  at  40%  below 
list  price.     What  per  cent  was  gained  ? 

76.  After  a  discount  of  30%  had  been  taken  off,  the  net 
price  of  some  goods  was  $  140.     What  was  the  list  price  ? 

77.  A  merchant  was  offered  a  bill  of  goods  for  $  1000,  with 
10%,  15%,  and  5%  off  for  cash.    He  offered  $  800  cash  for  the 
goods,  which  was  accepted.     Who,  if  either,  was  the  loser,  the 
buyer  or  the  seller  ?     How  much  ? 

Solve  the  equations  : 

78.  9^-12  =  8^-5. 

79.  8^-24-4^  +  16  =  32. 

80.  21a-35-8z  +  14  =  2a  +  89. 


4        4 

82.  4  times  a  certain  number  +  5  times  the  number  +  10  = 
82.    What  is  the  number? 

83.  An  agent  sold  a  sewing  machine  for  $  50  and  sent  to  the 
company  $  10  less  than  4  times  the  amount  of  his  own  com- 
mission.   How  much  was  his  commission  ?     How  much  did  he 
send  to  the  company  ? 

84.  In  three  camps  there  were  4000  soldiers.     In  the  second 
camp  there  were  3  times  as  many  men  as  in  the  first  camp,  and 
in  the  third  camp  4  times  as  many  as  in  the  first  camp.     How 
many  soldiers  in  each  camp  ? 

85.  The  perimeter  of  an  isosceles  triangle  is  32  inches.    Each 
of  the  equal  sides  is  7  inches  longer  than  the  base.    Find  length 
of  each  side. 

86.  Find  a  number  which  when  multiplied  by  7  and  divided 
by  8  gives  42  for  a  quotient. 


358  PROPORTION 

87.  The  perimeter  of  a  rectangle  is  110  cm.     Its  length  is  15 
cm.  greater  than  its  width.     Find  width  and  length.     How 
many  square  centimeters  in  each  of  the  triangles  formed  in  it 
by  its  diagonal  ? 

88.  Two  squares,  one  containing  144  sq.  in.  and  the  other 
121  sq.  in.,  are  placed  side  by  side  so  that  their  base  lines  form 
one  continuous  line.     Represent  and  find  perimeter  of  the  sur- 
face which  they  cover. 

89.  Divide  39  into  two  parts  such  that  one  part  shall  be  12 
times  the  other. 

90.  John  caught  three  times  as  many  fish  as  James.     Wil- 
liam caught  twice  as  many  as  John  and  James  both  caught. 
They  all  caught  120  fish.     How  many  fish  did  each  catch  ? 

91.  Tom,  Fred,  and  Will  whitewashed  both  sides  of  the  fence 
around  a  circular  lot  60  ft.  in  circumference.    Fred  whitewashed 
three  times  as  much  as  Tom,  Will  whitewashed  four  times  as 
much  as  Tom.     How  many  feet  of  the  fence  did  each  white- 
wash ? 

92.  The  circumference  of  a  given  circle  is  20  in.     How  long 
is  the  circumference  of  a  circle  whose  radius  is  4  times  as  long 
as  the  radius  of  the  given  circle  ? 

93.  A  merchant   bought  dress  goods  at  60^  a  yard   and 
marked  them  to  be  sold  at  an  advance  of  33 \%.     They  were 
sold  at  a  reduction  of  12£%  from  the  marked  price.    What  was 
the  selling  price  ?     What  per  cent  was  gained  ? 

94.  CLASS  EXERCISE.      may  name  a  cost  price  for  goods, 

a  marking  price,  and  a  selling  price,  which  is  a  certain  per  cent 
of  reduction  from  the  marked  price.      The  class  may  find  the 
per  cent  of  gain  or  loss  on  the  goods. 

95.  The   marked  price   of  some  silk  was  $  1.60  per  yard. 
It  was  sold  at  a  reduction  of  5%  from  the  marked  price.     The 
selling  price  was  27  ^  more  than  the  cost  price.    What  was  the 
per  cent  of  gain  ? 


MISCELLANEOUS  EXERCISES  359 

96.  When  lace  marked  $  1.50  per  yard  was  sold  at  a  reduc- 
tion of  16f  %  from  the  marked  price,  a  gain  of  $  .25  per  yard 
was  made.     Find  cost  price  and  per  cent  of  gain. 

97.  A  coat  marked  at  $  7.50  was  sold  at  a  reduction  of 
33^%.     The  selling  price  was  125%  of  the  cost  price.     Find 
cost. 

98.  A  suit  of  clothes  marked  $  25  was  sold  at  a  reduction 
of  20%  from  the  marked  price.     If  the  selling  price  was  25% 
above  the  cost  price  what  was  the  cost  price  ? 

99.  Goods  marked  at  40  ^  per  yard  were  sold  at  a  reduction 
of  12 £%  from  the  marked  price.     The  selling  price  was  75% 
above  cost.     Find  the  cost. 

100.  Some  goods  were  sold  for  $  .80,  which  was  25%  above 
the  cost  price.     How  much  did  they  cost  ?     The  selling  price 
was  20%  below  the  marking  price.     How  were  they  marked  ? 

101 .  Find  the  cost  and  the  marked  price  of  goods  sold  at  $  .75, 
which  was  a  reduction  of  25%  from  their  marked  price  and  an 
advance  of  50%  upon  their  cost. 

102.  Find  the  cost  and  the  marked  price  of  goods  sold  at 
$  1.20,  the  selling  price  being  an  advance  of  25%  upon  cost 
and  a  reduction  of  25%  from  the  marked  price. 

103.  CLASS  EXERCISE.     may  suppose  himself  to  be  a 

merchant  buying  goods  at  a  certain  price  and  marking  them 
to  sell  at  any  price  he  may  select.     Let  him  lower  the  marked 
price  by  a  certain  per  cent  and  then  find  the  per  cent  of  gain  or 
loss  which  the  selling  price  is  on  cost  price.     Let  him  give  the 
selling  price  to  the  class  and  tell  them  what  per  cent  that  is 
below  the  marked  price  and  above  or  below  the  cost  price. 
Let  the  class  find  the  marked  price  and  the  cost  price. 

104.  How  shall  goods  that  cost  80  ^  per  yard  be  marked  that 
a  reduction  of  10%  from  the  marked  price  may  be  made  and 
the  goods  sold  at  a  gain  of 


360  PROPORTION 

SOLUTION.  Let  x  =  the  number  of  dollars  in  the  marked  price.  Since 
a  reduction  of  10  %  from  the  marked  price  is  to  be  made,  the  actual  selling 
price  is  .90  x.  The  actual  selling  price  is  80^  +  12£%  of  80^,  which  is 
90.  Therefore 


Clearing  of  fractions,  90  x  — 

x  =  $  1.00 
Prove  this  and  the  following  problems. 

105.  I  marked  goods  which  cost  me  $  1.44  per  yard  so  that 
I  could  deduct  10%  from  the  marked  price  and  still  make  15% 
profit.     What  was  the  marked  price  ? 

106.  How  shall  I  mark  goods  that  cost  $  72,  so  as  to  deduct 
10%  from  the  marking  price  and  yet  gain  12-J-%  ? 

107.  A  buys  goods  for  $  12.     He  wishes  to  make  33^%  after 
discounting  20%  from  the  marked  price.     How  shall  he  mark 
them? 

108.  How  shall  goods  costing  75^  be  marked  that  10%  may 
be  deducted  from  the  marked  price  and  the  goods  still  be  sold 
at  a  profit  of  20%  ? 

109.  How  shall  goods  that  cost  $  1.20  be  marked  that  a  dis- 
count of  25%  from  the  marked  price  may  be  made  and  the 
goods  sold  at  a  profit  of  25%  ? 

110.  After  a  merchant  had  marked  goods  that  cost  90^,  a 
clerk  by  mistake  sold  them  at  a  reduction  of  40%   from  the 
marked  price.     This  selling  price  was  20%  below  cost.     What 
was  the  marked  price  ? 

111.  The  list  price  of  some  goods  is  $1.60.     A  merchant 
buys  them  at  a  discount  of  40%,  marks  them  to  be  sold  at  a 
profit  of  25%  on  the  cost  price,  and  discounts  them  to  the  cus- 
tomer 10%  from  marked  price.     How  much  does  the  merchant 
gain  ?     What  per  cent  ? 

112.  The  list  price  of  granite  soup  kettles  is  $  1.20  and  the 
discounts  are  50%  and  30%.     If  the  merchant  who  buys  them 


MISCELLANEOUS  EXERCISES  361 

marks  them  at  an  advance  of  16f  %  on  list  price,  and  dis- 
counts them  to  his  customer  40%  from  marked  price,  how 
much  does  he  gain  on  each  ?  What  per  cent  ? 

113.  A  merchant  buys  knives  at  a  discount  of  80%  and  10% 
from  list  price,  which  is  $  18  per  dozen.     He  marks  them  at  a 
price  40%  below  the  list  price,  and  sells  a  knife  to  a-  boy  at  a 
reduction  of  10%   from  the  marked  price.     How  much  and 
what  per  cent  does  he  gain  on  that  sale  ? 

114.  Find  the  length  of  the  hypotenuse  of  a  right  triangle 
whose  base  is  40  ft.  and  altitude  75  ft. 

115.  The  hypotenuse  of  a  right  triangle  whose  base  is  40 
ft.  and  altitude  42  ft.  is  how  much  less  than  the  sum  of  the 
other  two  sides  ? 

116.  If  the  hypotenuse  of  a  right  triangle  is  95  ft.  and  alti- 
tude 57  ft.,  how  long  is  the  base  ? 

117.  How  long  is  the  perimeter  of  a  right  triangle  whose 
base  is  42  ft.  and  hypotenuse  150  ft.  ? 

118.  How   long  is   the   longest   stick   that  can  be  carried 
through  a  doorway  6  ft.  high  and  2£  ft.  wide,  the  stick  being 
sharply  pointed  at  both  ends  ? 

119.  The  sum  of  all  the  edges  of  a  cube  was  132  in.     What 
was  the  volume  of  the  cube  ? 

120.  What  is  the  volume  of  a  cube  if  the  area  of  all  its 
faces  is  384  sq.  in.  ? 

Find  the  missing  term  in  each  of  the  following  proportions : 

121.  6:8  =  54:x.  126.    |:J  =  40:aj. 

122.  5:3  =  aj:21.  127.    2£  :  17£  =  3  :  x. 

123.  7:»  =  35:45.  128.    8J  :  35  =  x  :  20. 

124.  a;:  61  =  2: 122.  129.    3f:75  =  x:24. 

125.  i:f  =  6:o?.  130.    -J r:  £  of  f  =  42  :  x. 


362  PROPORTION 

Solve  by  proportion  and  by  analysis : 

131.  If  7  T.  of  hay  cost  $  77,  how  much  will  5  T.  cost  ? 

132.  If  10  men  earn  $  25  in  one  day,  how  much  will  3  men 
and  a  boy  earn  provided  the  boy  receives  half  as  much  as  a 
man? 

133.  If  a  man  travels  124  mi.  in  4  da.,  how  far  will  he  travel 
in  9  da.  ? 

134.  The  perimeter  of  a  right  triangle  is  8  ft.     The  hypote- 
nuse is  ^  ft.  4  in.  and  the  base  2  ft.  8  in.     Find  the  sides  of  a 
similar  triangle  whose  base  is  3  ft.  8  in. 

135.  The  circumference  of  a  given  circle  is  12.78  in.     Find 
the  circumference  of  a  circle  the  radius  of  which  is  ^  that  of 
the  given  circle. 

136.  If  8  bbl.  of  flour  can  be  made  from  40  bu.  of  wheat, 
how  many  barrels  of  flour  can  be  made  from  70  bu.  of  wheat  ? 

137.  If  75  United  States  bonds  can  be  bought  for  $  7725, 
how  much  will  60  of  the  same  bonds  cost?     At  3%  what  will 
be  the  income  from  the  60  bonds  ? 

138.  If  the  grocer's  bill  for  a  family  of  5  persons  is  $  15  per 
week,  what  will  be  that  of  a  family  of  7  persons  at  the  same 
rate  ? 

139.  $  40  pays  the  board  of  10  persons  for  a  week  at  a  farm- 
house.    If  the  price  of  board  were  doubled,  how  many  persons 
could  obtain  board  for  the  same  time  for  the  same  amount  ? 

140.  If  15  men  can  do  a  piece  of  work  in  40  da.,  how  long 
will  it  take  12  men  to  do  it  ? 

141.  If  30  men  can  do  a  piece  of  work  in  16  da.,  how  many 
men  can  do  the  same  work  in  15  da.  ? 

142.  Mr.  A.  holds  stock  in  a  company  which  last  year  paid 
12%  dividends.     His  dividends  were  $  192.     If  this  year  the 
rate  of  dividend  is  9%,  how  much  will  he  receive  from  that 
investment  ? 


MISCELLANEOUS  EXERCISES  363 

143.  Mrs.  C.  has  a  yearly  income  of  $  1876  from  an  invest- 
ment which  pays  8%.     If  it  paid  6%,  how  much  would  she 
receive  from  that  investment  ? 

144.  When  Mr.  D.'s  house  is  rented  at  $  40  a  month  the  rent 
is  8%  of  the  cost  of  the  house.     What  per  cent  of  the  cost  of 
the  house  does  he  receive  when  the  tenant  pays  $  35  a  month  ? 

145.  A   quantity  of  wheat  was  shipped  1400  mi.     For  the 
first  800  mi.  the  charges  were  $  60.    For  the  rest  of  the  distance 
the  rate  per  mile  was  twice  as  great.     What  was  the  cost  of 
transportation  for  the  whole  distance  ? 

146.  How  would  you  divide  $  90  among  three  persons  in  the 
ratios  of  2,  3,  and  4  ? 

147.  A  and  B  formed  a  partnership,  A's  capital  being  ^  as 
much  as  B's.     Their  profits  were  $  8000  the  first  year.     How 
much  ought  each  to  receive  ? 

148.  Mr.  Ball  owned  twice  as  many  shares  in  a  mining  com- 
pany as  his  brother.     The  sum  of  their  dividends  was  $  1800. 
How  much  dividend  should  each  receive  ? 

149.  Three   heirs,  James,  Lucy,  and  Henry,  own   a  farm 
which  rents  for  $  2100  a  year.     James  owns  3  times  as  many 
acres  as  Lucy,  and  Lucy  owns  twice  as  many  as  Henry.     How 
shall  the  rent  be  divided  ? 

150.  The  distance  from  Alta  to  Eeed's  Crossing  is  84  mi.  by 
the  A.  &  M.  E.E.,  and  from  Eeed's  Crossing  to  Doane  96  mi. 
by  the  C.  &  D.  E.E.     The  freight  charges  on  some  merchandise 
carried  from  Alta  to  Doane  are  $  29.16.     How  much  shall  each 
railroad  receive  from  it  ? 

151.  The  freight  charge  on  merchandise  shipped  824  mi. 
over  three  railroad  lines  is  f  206.     To  how  much  is  the  A.  &  X. 
B.K.  entitled  which  carried  it  378  mi.  ?    What  sum  belongs  to 
the  C.  &  Y.  E.E.  which  carried  it  212  mi.  ?     To  the  third 
railroad  ? 


364  PROPORTION 

At  5%  what  is  the  interest  of 

152.  $  725  from  May  9,  1892,  to  Sept.  11,  1899  ? 

153.  $638  from  June  17,  1873,  to  May  21,  1880  ? 

154.  $525  from  Aug.  12,  1884,  to  June  17,  1893? 

155.  $631  from  Jan.  21,  1887,  to  June  20,  1892  ? 

156.  $375  from  Oct.  13,  1899,  to  Feb.  3,  1905? 

157.  How  long  is  the  side  of  a  square  whose  area  is  equal 
to  that  of  a  rectangle  36  in.  long  and  4  in.  wide  ?     The  perim- 
eter of  the  square  equals  what  per  cent  of  the  perimeter  of 
the  oblong  ? 

158.  How  long  is  the  side  of  a  square  whose  area  is  equal 
to  that  of  a  rectangle  whose  length  is  343  ft.  and  width  7  ft.  ? 
The  perimeter  of  the  square  equals  what  per  cent  of  the  perim- 
eter of  the  oblong  ? 

159.  A  agreed  to  dig  potatoes  for  B,  taking  £  of  the  potatoes 
for  his  pay.     He  dug  5  bu.  of  potatoes  and  set  them  aside  for 
his  employer.     The  next  bushel  he  set  aside  for  himself,  and 
so  continued.     When  he  had  dug  600  bu.,  how  many  bushels 
that  belonged  to  him  had  he  failed  to  get   because   of  his 
ignorance  of  ratio? 

160.  Two  paper  hangers  finished  papering  a  house.     They 
were  to  receive  $  22.60  for  their  work.     The  first  man  was  to 
receive  $1.40  more  than  the  other;  they  divided  the  $22.60 
into  two  equal  parts ;  then  the  second  man  gave  the  first  man 
$1.40.     Was   the   division   right?     If  not,   how   may  it   be 
corrected  ? 

161.  Find  the  sum  of  the  squares  of  two  numbers,  as  9  and 
7.     Find  the  difference  of  their  squares.     Add  the  difference 
of  their  squares  to  the  sum  of  their  squares.     Divide  the  num- 
ber thus  found  by  2,  extract  the  square  root,  and  the  result 
will  be  the  greater  number.     Take  several  pairs  of  unequal 
numbers  and  see  if  this  holds  true. 


MISCELLANEOUS  EXERCISES  365 


162.  VF+2^?       Vl3+23+33=?      Vl3+23+33+43=? 

163.  Find  the  square  root  of  the  sum  of  the  first  5  perfect 
cubes.     Of  the  first  6  perfect  cubes. 

164.  Mr.  Wilson  is  building  a  house  which  has  a  roof  45  ft. 
long  with  rafters  15   ft.  long.     How   many  shingles   will  be 
used  to  cover  the  two  sides  of  the  roof  if  900  shingles  4  in. 
wide  and  exposed  4J  in.  to  the  weather  are  required  to  cover 
1  square  10  ft.  in  dimensions  ?     How  much  will  the  shingles 
cost  at  $2.75  a  thousand?      At  3^  per  pound,  how  much 
will  the  shingle  nails  cost  if  5  Ib.  are  required  for  each  square  ? 
How  much  must  be  paid  to  the  carpenter  for  shingling  the  roof 
at  $  1.25  per  square  ?     What  is  the  entire  cost  of  shingling  the 
roof? 

165.  Think  of  a  number,  multiply  it  by  6,  add  9,  divide  the 
result  by  3,  subtract  3,  divide  the  remainder  by  2,  and  you  will 
have  the  original  number.     Try  this  with  several  numbers,  and 
then  try  to  find  why  this  is  true  whatever  the  original  number 
may  be. 

166.  What  is  the  ratio  of  the  perimeter  of  an  equilateral 
triangle  a  side  of  which  is  3  in.  to  the  perimeter  of  an  equi- 
lateral triangle  a  side  of  which  is  6  in.  ? 

167.  Take  Ex.  166  substituting  "hexagon"  for  " triangle." 

168.  Take  Ex.  166  substituting  "pentagon"  for  "triangle." 

169.  How  many  right  angles  can  a  trapezium  have?     A 
trapezoid  ?     A  rhomboid  ? 


CHAPTER   XII 

MEASUREMENTS  AND  CONSTRUCTIONS 
LINES  AND  SURFACES 

1.  A   plane   figure   bounded  by  straight  lines  is  called   a 
Polygon. 

Draw  a  polygon  of  five  sides. 

A  polygon  of  three  sides  is  a  Triangle  ;  of  four  sides,  a  Quadrilateral ; 
of  five  sides,  a  Pentagon  ;  of  six  sides,  a  Hexagon ;  of  seven  sides,  a  Hep- 
tagon ;  of  eight  sides,  an  Octagon ;  of  nine  sides,  a  Nonagon ;  of  ten  sides, 
a  Decagon ;  of  twelve  sides,  a  Dodecagon. 

2.  A  polygon  having  all  its  sides  equal  and  all  its  angles 
equal  is  called  a  Regular  Polygon. 

Draw  a  regular  polygon. 

Polygons  which  approximate  very  closely  to  regular  polygons  may  be 
constructed  by  a  method  of  which  the  following 
construction  of  an  approximately  regular  heptagon 
is  an  illustration  :  Draw  AB  7  units  long,  and 
mark  the  divisions  of  units.  Draw  a  circle  of 
which  AB  is  the  diameter.  With  A  as  a  center 
and  AB  as  a  radius  draw  an  arc.  With  the  same 
radius  and  with  B  as  a  center  draw  an  arc  inter- 
secting the  first  arc  at  C.  Draw  (72  passing 
through  the  second  division  of  the  diameter, 
and  prolong  it  until  it  meets  the  circumference 
at  D.  AD  will  lie  seven  times  consecutively  as  a 
chord. 

366 


LINES  AND   SURFACES  367 

A  polygon  of  any  required  number  of  sides  may  be  constructed  in  the 
same  way  by  making  the  number  of  divisions  in  the  diameter  equal  to 
the  required  number  of  sides.  The  line  CD  must  always  pass  through 
the  second  division  of  the  diameter. 

3.  Draw  a  heptagon  by  the  method  described  in  the  note 
and  change  it  to  a  seven-pointed  star.     In  the  same  way  draw 
a  five-pointed  star  and  a  nine-pointed  star. 

4.  Triangles  are  classified   with  regard  to   their  sides  as, 
Equilateral,  having  three  equal  sides;   Isosceles,  having  two 
equal  sides ;  and  Scalene,  having  no  two  sides  equal. 

Construct  an  equilateral  triangle,  an  isosceles  triangle,  and 
a  scalene  triangle,  each  having  a  perimeter  of  15  in. 

5.  Triangles  are  classified  with  regard  to  their  angles  as  Right 
triangles,  Obtuse-angled  triangles,  and  Acute-angled  triangles. 

Draw  a  triangle  containing  a  right  angle. 

A 

6.  A  Right  triangle  is  one  that  has  a 
right  angle. 

Draw  a  scalene  right  triangle.  An 
isosceles  right  triangle. 


7.  An  Obtuse-angled  triangle  is  a  triangle 
that  has  an  obtuse  angle. 

Draw  a  scalene  obtuse-angled  triangle.  An 
isosceles  obtuse-angled  triangle. 


8.  An  Acute-angled  triangle  is  a  triangle 
in  which  all  the  angles  are  acute. 

Draw  a  scalene  obtuse-angled  triangle.  An 
isosceles  acute-angled  triangle. 


9.    Into  what  kind  of  triangles  is  a  rhombus  divided  by  its 
long  diagonal  ?     By  its  short  diagonal  ? 


368 


MEASUREMENTS  AND   CONSTRUCTIONS 


10.    Quadrilaterals  are  of  three  kinds,  trapeziums,  trapezoids, 
and  parallelograms : 


Trapezium.     No  parallel 
sides. 


Trapezoid.    Two  parallel 
sides. 


Quadri- 
laterals. 


Parallelo- 
gram. 
Opposite 
sides  par- 
allel. 


Kectangle. 
All  angles 
right  angles. 


Square.     Equilateral 
rectangle. 


Rhomboid. 
Angles  oblique. 


Ehombus.     Equilateral 
rhomboid. 


Draw  four  kinds  of  parallelograms  and  write  the  name  of 
each  upon  it. 

11.  State  the  difference  between  a  square  and  any  other 
rectangle.  Between  a  rectangle  and  a  rhomboid.  Between  a 
square  and  a  rhombus.  Between  a  rhombus  and  any  other 
rhomboid.  Between  a  trapezoid  and  a  trapezium.  Between 
a  trapezium  and  a  rectangle. 


LINES   AND   SURFACES  369 

• 

12.  Find  the  area  of  a  rectangle  whose  base  is  7  in.  and  alti- 
tude 4  in.     Find  the  area  of  a  right  triangle  having  the  same 
base  and  altitude. 

13.  Represent  a  trapezoid  whose  lower  base  is  8  in.  and 
upper  base  6  in.,  the  bases  being  4  in.  apart.     Find  the  area  of 
the  trapezoid. 

14.  The  parallel  sides  of  a  field  of  trapezoidal  shape  are 
120  rd.  and  80  rd.  long;  they  are  90  rd.  apart.     How  many 
acres  are  there  in  the  field  ? 

15.  What  is  the  altitude  of  a  trapezoid  whose  area  is  825 
sq.  ft.  and  whose  bases  are  60  and  90  ft.  ? 

Let  x  =  the  number  of  feet  in  the  altitude. 

16.  Which  is  greater,  and  how  much,  a  trapezoid  whose 
parallel  sides  are  10  in.   and  4  in.,  and  altitude  5  in.,  or  a 
trapezoid  of  the  same  altitude  whose  parallel  sides  are  8  in. 
and  6  in.  ?     Represent. 

17.  The  sum  of  the  parallel  sides  of  a  trapezoid  is  16  in.  and 

the  altitude  is  3  in.    What  is  the  area  ? 

18.  Draw  and  cut  out  the  rhom- 
boid ABCD.  Draw  AE  perpendicu- 
lar to  DC  (Fig.  1).  Cut  off  the  tri- 
angle  ADE  and  place  it  on  the  other 
side  of  the  rhomboid  so  that  AD  and 
BC  unite  (Fig.  2).  The  figure  thus 
formed  will  be  a  rectangle  having  the 
same  base  and  altitude  as  the  rhom- 
boid. If  the  base  DC  of  the  rhomboid 
is  7  in.  and  altitude  AE  3  in.,  what  is 
the  area  of  the  rhomboid  ? 

19.  Draw  and  cut  out  a  rectangle  6  in.  by  4  in.  Cut  and 
arrange  its  parts  into  a  rhomboid  of  equal  area.  If  you  had 
another  rectangle  6  in.  by  4  in.,  could  you  make  another  rhom- 
boid equal  to  the  first  but  of  different  shape  ?  Explain. 

HORN.    GRAM.    SCH.    AR. — 24 


370 


MEASUREMENTS   AND   CONSTRUCTIONS 


20.  Make  problems  to  illustrate  the  following  rule : 
To  find  the  area  of  a  rhomboid  — 

Multiply  the  base  by  the  altitude. 

21.  Find  the  area  of  a  rhomboid  whose  longer  sides  are  9  in. 
apart,  and  are  each  48  in.  long. 

Supply  the  missing  numbers  in  the  measurements  of  the 
following  rhomboids : 

Alt. 

5  in. 

10  in. 


Base 

22. 

40  in. 

23. 

x  in. 

24. 

25  in. 

25. 

13.8  in. 

26. 

x  in. 

27. 

5ft. 

Area 

x  sq.  in. 
60  sq.  in. 
175  sq.  in. 


55.2  sq.  in. 
10.2  sq.  in. 

2  sq.  ft.  12  sq.  in. 


x  in. 
x  in. 
251  in. 
x  in. 

28.  The  perimeter  of  a  certain  rhomboid  is  60  in.    The  base 
is  20  in.     The  altitude  is  6  in.     Represent  and  find  area. 

29.  A  base  of  a  rhomboid  is  5  in.  longer  than  the  altitude, 
which  is  16  in.     What  is  the  area  of  the  rhomboid  ? 

30.  The  altitude  of  a  rhomboid  is  sometimes 
represented  by  a  line  that  falls  outside  the  rhom- 
boid. BE,  the  altitude,  is  the  perpendicular  dis- 
tance between  the  side  AB  and  the  opposite  side 
produced.  If  DC  is  7  in.  and  BE  8  in.,  what  is 
the  area  of  the  rhomboid  ABCD  ? 

31.  LB  and   ED  are 

parallel.  Which  is  the 
greater  rhomboid,  OFKL 
or  GFBC?  Explain. 

32.  Reproduce  LB  and 
ED,  making  them  5  in. 
apart.     Let  GF  be  4  in. 

Draw  several  rhomboids  whose  base  is   GF  and  whose  side 
parallel  to  the  base  is  a  part  of  LB.     What  is  the  area  of  each  ? 


K 


F 
FIG.  4. 


LINES  AND   SURFACES  371 

33.  A  farmer  who  had  a  rectangular  lot  300  ft,  long  and 
80  ft.  wide,  sold  a  strip  of  it  to  a  railway  company  for  a  right 
of  way.  The  agreement  was  that,  beginning  at  the  south- 
west corner  of  his  lot,  the  width  of  the  strip  should  be 
measured  off  80  ft.  along  the  southern  boundary  of  the  lot, 
that  the  strip  should  thence  cross  the  lot,  bounded  by  straight 
parallel  lines,  and  that  the  railroad  company  should  pay  5  ^  a 
square  foot  for  it.  How  much  did  the  farmer  receive  for  the 
land? 

34.    The  shaded  part  of  Fig.  5  shows  the  portion  of  land 
that  the  farmer  intended  to  sell.     The  shaded  part  of  Fig.  6 
E    shows  the  portion  of   land  that  the 
company  bought.     AD  being  80  lit.,  is 
the  strip  in  accordance  with  the  agree- 
ment ? 

35.  The  farmer,  thinking  that  the 
company  had  taken  more  land  than 
belonged  to  it,  consulted  a  lawyer, 
who  proved  to  him  that  the  two  strips 
were  equal,  by  drawing  diagrams  like 
Fig.  5  and  Fig.  6,  and  showing  him 
that  if  the  two  triangles  AHB  and  DCG  in  Fig.  6  were  cut 
out  and  applied  to  the  rectangle  CEFD  in  Fig.  5,  they  would 
exactly  equal  it.  How  could  you  determine  whether  or  not 
the  strips  were  equal  ? 

36.  Construct  a  rhomboid,  cut  and  rearrange  its  parts  in 
such  a  way  as  to  make  a  trapezoid  of  equal  area.     A  rectangle 
of  equal  area. 

37.  What  name  is  given  to  a  rectangle  whose   base  and 
altitude  are  equal  ? 

38.  Construct  a  rhomboid  whose  base  and  altitude  are  each 
3  in.     Cut  and  rearrange  its  parts  into  a  square. 


372 


MEASUREMENTS   AND   CONSTRUCTIONS 


39.  Find  the  area  of  the  rhom- 
bus ABCD  if  AB  is  8  in.  and  EB 

7  in. 

40.  Find  the  area  of  a  rhombus 
whose  perimeter  is  4  ft.  4  in.,  and 
whose  altitude  is  6  in.  less  than 

FIG.  7.  one  side. 

41.  Find  the  area  of  a  rhombus  whose  perimeter  is  40  in. 
and  whose  altitude  is  f  of  a  side. 

42.  Is  a  rhombus  a  regular  polygon  ?      Give  reasons  for 
your  answer. 

43.  Draw  a  square  whose  sides  are  each  2  in.  long,  and  a 
rhombus  whose  sides  are  each  2  in.  long.      Which   has  the 
greater  altitude?     The  greater  area? 

44.  Can  you  draw  a  rhombus  in  which  the  base  and  altitude 
are  equal  ?     Explain. 

45.  If  the  rectangle  AEDC  were   8 
in.  long  and  5  in.  wide,  what  would  be 
the  area  of  the  triangle  ABC?     Give 
reasons. 

46.  What  would  be  the  area  of  ABC 
if  AEDC  were    9    in.  long  and  4  in. 
wide?      50  in.  long  and  18  in.  wide? 

6  ft.  3  in.  long  and  1  ft.  wide? 

47.  Draw    a   figure   and  show  the  truth  of  the  following 
statement: 

The  area  of  a  triangle  is  equal  to  one  half  the  product  of  its 
base  and  altitude. 

48.  Find  the  area  of  a  triangle  whose  base  is  18  in.  and 
altitude  5  in. 

49.  Find  the  area  of  a  triangle  whose  base  is  17  in.,  and 
whose  altitude  is  5  in.  greater  than  the  base. 

50.  What  is  the  area  of  a  triangle  whose  base  is  2  ft.  6  in., 
and  altitude     as  much? 


H 
FIG.  8. 


LINES  AND  SURFACES 


373 


51.    If  AD  were  10  in.,  and  CE  5 
in.,  what  would  be  the   area   of  the 
Of  the  triangle 


rhomboid  ABCD? 
ADC ?     ABC ? 


FIG.  9. 


52.    If  AD  were  24  cm.,  and  CE 
were  half  as  long  as  AD,  how  many 
would    the    rhomboid     contain?      Each 


square  '  centimeters 
triangle  ? 

53.  Either  side  of  a  triangle  may  be  considered  a  base.  The 
angle  opposite  the  base  is  called  the  Vertical  Angle.  The 
Altitude  is  the  perpendicular  distance  from  the  vertical  angle 
to  the  line  of  the  base. 

B  is  a  right  angle.     If  AC  is 

B 

considered  the  base,  what  angle 
is  the  vertical  angle?  What  line 
is  the  altitude  ?  If  AB  is  the  base, 
what  angle  is  the  vertical  angle? 
What  line  is  the  altitude?  If  AB 
is  6  in.,  and  BC  is  10  in.,  what  is 

Which  ^B  x 


Fio.  10. 


the   area  of  the  triangle? 
AC  x  BD  0 


is  greater, 


or 


The  altitude  of  a  triangle  is  sometimes  represented  by  a  line  that  falls 
outside  the  triangle. 

54.  If    BC  is  considered  the   base  of 
the  triangle  ABC,  what  line  is  the  alti- 
tude?     If  BC  is  1  in.,  and   AD  6  in., 
what  is  the  area  of  the  triangle  ? 

55.  Reproduce  the  triangle  ABC,  and 
draw  a  line  to  represent  the  altitude  when 
BC  is  considered  the  base.     When  AB  is 

the  base.     When  CB  is  the  base. 

56.  If  CB  were  10  in.,  and  the  corresponding  altitude  6  in., 
what  would  be  the  area  of  the  triangle  ?  If  AB  were  12  in., 
how  long  would  the  corresponding  altitude  be  ? 


FIG.  ii. 


374 


MEASUREMENTS   AND   CONSTRUCTIONS 


58. 
59. 
60. 
61. 
62. 

Base 
10  in. 
6  in. 
20ft. 
60yd. 
15  rd. 

Alt. 
Tin. 
9  in. 
30ft. 
40yd. 
13rd. 

63. 
64. 
65. 
66. 
67. 

Base 
12ft. 
13ft. 
x  in. 
x  ft. 
20  rd. 

Alt. 
X  ft. 

£ft. 

10  in. 
50ft. 
x  rd. 

36 
65 
200 
750 
300 

Area 
sq.  ft. 
sq.  ft. 
sq.  in. 
sq.  ft. 
sq.  rd. 

57.  The  longest  side  of  a  triangular  field  is  120  rd.  The 
perpendicular  distance  from  the  opposite  corner  to  that  side 
equals  33J%  of  the  side.  Find  the  area  of  the  field  in  square 
rods.  In  acres. 

Find  the  missing  measurements  of  the  following  triangles  : 

Area 

x  sq.  in. 
x  sq.  in. 
x  sq.  ft. 
x  sq.  yd. 
x  sq.  rd. 

68.  The  base  of  a  triangle  is  45  in.,  and  the  altitude  is  twice 
the  base.     What  is  the  area? 

69.  The  altitude  of  a  triangle  is  14  in.,  and  the  base  is  3 
times  the  altitude.     What  is  the  area? 

70.  Find  the  area  of  a  triangle  whose  base  is  8  in.  and  alti- 
tude 9  in.     How  long  is  the  side  of  a  square  whose  area  equals 
that  of  the  triangle  ? 

71.  Find  the  perimeter  of  a  square  equal  in  area  to  a  tri- 
angle whose  base  is  20  in.,  and  altitude  10  in. 

72.  How  wide  is  a  rectangle  18  in.  long,  and  equal  in  area  to 
a  triangle  whose  base  is  12  in.  and  altitude  9  in.  ? 

73.  Draw  an  isosceles  right  triangle.      If  each  of  the  equal 
sides  were  12  in.  long,  what  would  be  its  area?     If  a  rectangle 
equal  in  area  to  the  triangle  is  3  in.  wide,  what  is  its  length? 

74.  By  folding  an  isosceles  triangle  in 
such  a  way  that  the  equal  sides  coincide,  it 
will  be  seen  that  a  line  drawn  from  the  ver- 
tical angle  to  the  middle  of  the  base  divides 
the  triangle  into  two  equal  right  triangles. 
How  long  is  the  altitude  AD  of  the  isosceles 
triangle,  of  which  the  base  CB  is  10  in.,  and 
the  equal  sides  are  each  13  in.  ? 


FIG.  12'. 


LINES   AND   SURFACES  375 

75.  Find  the  altitude  of  an  isosceles  triangle  whose  base 
is  30  in.,  and  whose  equal  sides  are  each  39  in.    Find  the  area  of 
the  triangle. 

76.  Find  the  altitude  of  an  isosceles  triangle  whose  perim- 
eter is  50  in.  and  base  16  in.     Find  the  area. 

77.  The  area  of  an  isosceles  triangle  whose  base  is  18  in. 
is  108  sq.  in.     What  is  its  altitude  ?     The  length  of  one  of  its 
equal  sides  ?     Its  perimeter  ? 

78.  Given  an   isosceles  triangle  whose  base  is  40  in.  and 
area  300  sq.   in.     Find  its   altitude.     Find  one  of  its  equal 
sides.     Find  its  perimeter. 

79.  Given  an  isosceles  triangle  whose  altitude  is  42  in.  and 
area  1680  sq.  in.     Find  the  base.     Find  the  length  of  each  of 
the  equal  sides. 

80.  Find  to  one  place  of  decimals  the  altitude  of  an  equilat- 
eral triangle  whose  side  is  10  in.     Find  its  area. 

81.  Find  the  altitude  and  the  area  of  an  equilateral  triangle 
whose  side  is  8  in. 

82.  Find  the  area  of  an  equilateral  triangle  whose  perimeter 
is  60  in. 

83.  Arrange  6  equilateral  triangles  so  as  to  form  a  regular 
hexagon,  and  find  the  area  of   the  hexagon,  supposing  each 
side  of  the  triangles  to  be  12  in. 

84.  Construct  a  regular  octagon,  and  draw  a  line  from  its 
center  to  the  vertex  of  each  angle. 

85.  Into  how  many  and  what  kind  of  triangles  is  a  regular 
octagon  divided  by  lines  drawn  from  its  center  to  the  extremi- 
ties of  its  sides  ? 

86.  The  distance  from  the  center  of  a  regular  polygon  to  the 
middle  point  of  one  of  its  sides  is  called  the  Apothem  of  the 
polygon. 

Draw  a  regular  polygon  and  a  line  to  show  its  apothem. 


376 


MEASUREMENTS  AND   CONSTRUCTIONS 


87.    How  does  the  apothem  of  a  polygon  compare  with  the 
altitude  of  one  of  the  isosceles  triangles  into  which  a  regular 
polygon  may  be  divided  by  lines  drawn 
from  its  center  to  the  extremities  of  the 
sides  ? 

88.  If  a  side  of  the  regular  pentagon 
BCDEF  were  8  in.,  the  apothem  OA 
would  be  5.44  in.  What  would  be  the 
area  of  the  triangle  OBC?  Of  the 
whole  pentagon  ? 

89.  If  a  regular  pentagon  were  cut  into  5  equal  isosceles 
triangles,  and  arranged  as  in  Fig.  14,  the  sum  of  the  bases  of 
the  triangles  would  equal  what  line  ? 


E  D 

FIG.  13. 


FIG.  14. 

90.  Since  every  regular  polygon  may  be  divided  into  as 
many  isosceles  triangles  as  it  has  sides,  we  may  find  the  area 
of  a  regular  polygon  by  the  following  principle : 

The  area  of  a  regular  polygon  is  equal  to  one  half  the  prod- 
uct of  its  perimeter  and  apothem. 

What  is  the  area  of  a  regular  pentagon  if  the  perimeter-  is 
80  in.  and  the  apothem  10.88  in.  ? 

91.  The   ratio   of    the   apothem   to   the   side  of    a  regular 
pentagon  =  .68  ;   of  a  regular  heptagon  =  1.03 ;    of  a  regular 
octagon  =  1.20 ;  of  a  regular  decagon  =  1.86.     The  apothem  of 
a  regular  hexagon  can  be  easily  found  by  the  Pythagorean 
Theorem. 

If  the  perimeter  of  a  regular  pentagon  is  60  in.,  what  is  the 
apothem?  The  area? 

92.  Find  the  area  of  a  regular  decagon,  one  side  of  which  is 
8  in. 


LINES   AND   SUKFACES 


377 


Given  one  side  14  in. : 

93.  Find  the  area  of  a  regular  pentagon. 

94.  Of  a  regular  hexagon.        96.    Of  a  regular  decagon. 

95.  Of  a  regular  octagon.         97.    Of  a  regular  heptagon. 

98.  What  is  the  area  of  a  flower  bed  in  the  shape  of  a 
regular  hexagon  whose  perimeter  is  96  ft.? 

99.  At  $  1.25  per  square  foot,  what  is  the  value  of  a  park 
in  the  shape  of  a  regular  octagon,  each  side  of  which  is  40  ft.? 

100.  In  decorating  a  schoolroom  15  six-pointed  stars  were 
used.  Each  star  was  made  by  combining  12  triangles,  each 
side  of  which  was  4  in.  Draw  a  figure  to  show  how  the  stars 

were  made.  Find  the  cost  of 
gilding  all  of  them  at  5  ^  a  square 
foot. 

101.  Figure     15    represents    a 
"block"  of  patchwork  from  a  quilt. 
Find  the  area  of  the  whole  block 
if  the  perimeter  of  the  inner  hexa- 
gon is  12  in. 

102.  What     is    the     difference 
between    the    area    of    a    square 

whose  perimeter  is  24  in.  and  the  area  of  a  regular  hexagon 
whose  perimeter  is  24  in.  ? 

103.  Construct  the    regular    hexagon 
ABODE 'F,  each   side   being  4   in.,    and 
divide  it  into  equal  trapezoids  by  a  line 
represented  by  AD.     What   is  the  area 
of  each  trapezoid? 

104.  Draw  the  square  GHJK.     What 
is  its  area?     How  long  is  the  perimeter 
of  the  irregular  figure  ABCDJH?   What 
is  its  area? 


FIG.  15. 


378 


MEASUREMENTS   AND   CONSTRUCTIONS 


105.  If  the  apothem  of  a  regular  pentagon  is  20.4  in.,  how 
long  is  one  side? 

See  table  of  ratios,  p.  376. 

106.  Find  the  area  of  a  regular  decagon  whose  apothem  is 

B  130.2  in. 

107.  Find  the  area  of  a  regular 
octagon  whose  apothem  is  13.2  in. 

108.  Find  the  area  of  a  regular 
heptagon  whose  apothem  is  175.1 
in. 

109.  Find    the    area    of    the 
trapezium  ABCD,  if  the  line  AC 
is  8  in.,  BE  perpendicular  to  AC 
4  in.,  and  FD  perpendicular  to 
AC  3  in. 

In  finding  the  area  of  an  irregular  figure  it  is  customary  to  divide  it  into 
triangles  or  trapezoids  and  find  the  area  of  each  part  separately. 

110.  A  surveyor  found  the  area  of  a  piece  of  land  repre- 
sented by  the  irregular  pentagon  ABODE.  The  diagonal  AD 
was  50  ch.,  of  which  AH  was  10  ch.  and  HF  28  ch. 

o 


17. 


FIG.  18. 


The  perpendiculars  BH,  CF,  and  EG  were  respectively  12 
ch.,  18  ch.,  and  15  ch.  A  chain  measures  4  rd.  How  many 
acres  were  there  ? 

It  will  be  observed  that  HBCF  is  a  trapezoid  and  that  the  angles  at  H 
and  at  F  are  right  angles. 


LINES   AND   SURFACES  379 

111.    CLASS  EXERCISE.     may  draw  an  irregular  polygon 

upon  the  board,  and  the  class  may  show  different  ways  of  divid- 
ing it  into  triangles  or  trapezoids  to  find  its  area. 


112.  A  polygon  is  said  to  be  inscribed  in 
a  circle  when  the  vertex  of  each  angle  of 
the  polygon  is  in  the  circumference  of  the 
circle.  The  circle  is  said  to  be  circumscribed 
about  the  polygon. 

Inscribe  a  regular  hexagon  in  a  circle  as 
5*Ti9.  in  Fig.  19. 

113.  Can  a  rhombus  be  inscribed  in  a  circle  ?     Explain. 

114.  Can  a  hexagon  whose  sides  are  each  4  in.  long  be  drawn 
within  a  circle  whose  radius  is  5  in.  ?     Can  it  be  inscribed  in 
the  circle  ?     Explain. 

115.  If  the  side  of  a  regular  hexagon  is  5  in.,  how  long  is 
the  radius  of  the  circumscribed  circle  ?     How  long  is  its  cir- 
cumference ? 

116.  The  part  of  a  circle  between  an  arc  and  its  chord  is 
called  a  Segment. 

How  long  would  be  the  perimeter  of  each  segment  of  Fig.  19, 
if  the  radius  of  the  circle  were  24^-  in.  ? 

117.  Bisect  each  arc  of  your  copy  of 
Fig.  19.  Join  the  middle  point  of  each  arc 
with  the  extremities  of  its  chord  as  in 
Fig.  20.  How  many  sides  has  the  regular 
polygon  you  have  thus  formed  ?  How 
would  its  area  be  found  if  the  length  of 

one  side  and  the  apothem  were  known  ? 
FIG.  20. 

If  in  the  same  way  a  polygon  of  24  sides  were 

inscribed  in  the  circle,  and  then  another  of  double  that  number  of  sides, 
and  so  on,  we  should  soon  have  a  polygon  whose  sides  were  so  small  that 
the  perimeter  of  the  polygon  could  not  be  distinguished  from  the  circum- 
ference of  the  circle,  and  the  polygon  and  the  circle  would  appear  to  be 
the  same. 


880  MEASUREMENTS   AND   CONSTRUCTIONS 

118.    A  circle  may  be  considered  as  a  polygon  of  an  infinitely 
great  number  of  sides,  its  circumference  being  the  perimeter  of 
the  polygon.    To  what  would  the  radius  of  the  circle  correspond  ? 
119.    Cut  out  a  small  circle  and  fold 
it  in  halves.     Fold  it  again,  and  continue 
folding   until   many  small    sectors   are 
made  by  the  folds.     Cut  along  the  folds 
and  place  the  circle  as  in  Fig.  21. 

To  what  is  the  sum  of  the  bases  of 
these  sectors  equal  ?  To  what  is  their 
altitude  equal  ? 

120.  If  a  circle  is  considered  as  a  polygon  of  an  infinitely 
great  number  of  sides,  the  circumference  is  the  sum  of  those 
sides  and  the  radius  of  the  circle  is  the  apothem  of  the  poly- 
gon.    Can  you  see  the  reason  for  the  following  fact  ? 

The  area  of  a  circle  is  equal  to  one  half  the  product  of  its  cir- 
cumference and  radius. 

This  may  be  expressed  by  the  formula  A=C  x  -  or  A=  — ,  in  which 

"-4"  stands  for  "area  of  circle,"  "C"'  for  "circumference,"  and  "r" 
for  "radius." 

121.  Find  the  area  of  a  circle  whose  radius  is  7  ft. 
Find  the  areas  of  circles  of  the  following  dimensions : 

122.  Radius  10  ft.  125.    Circumference  77  ft. 

123.  Circumference  110  ft.     126.    Eadius  1  ft.  9  in. 

124.  Diameter  18  in.  127.    Circumference  5  ft.  10  in. 

128.  Measure  or  estimate  the  diameter  of  the  face  of  a  clock 
or  watch  and  find  its  area. 

129.  The  minute  hand  of  a  clock  in  a  tower  is  3  ft.  6  in. 
long.     What  is  the  area  of. that  part  of  the  clock  face  over 
which  the  exact  middle  line  of  the  hand  passes  in  15  min.  ? 

130.  What  is  the  area  of  a  sector  which  is  ^  of  a  circle,  if  the 
arc  of  the  sector  is  55  in.  ?     Represent. 

131.  A  cow  is  tied  to  a  stake  in  a  field  by  a  rope  20  ft.  long. 
What  is  the  area  of  the  surface  over  which  she  can  move  ? 


LINES  AND   SURFACES 


381 


132.  A  statue  whose  base  is  8  ft.  square  is  placed  in  the 
center  of  a  grass  plot  having  a  circumference  of  1254  ft.  How 
many  square  feet  of  the  grass  plot  are  around  the  statue  ? 

133.  AB,  a  diameter,  is  14  in.     How 
long  is  one  side  of  the  inscribed  square  ? 
What  is  its  area  ?     Find  the  area  of  each 
segment  cut  off  by  the  square.     Find  the 
length  of  its  perimeter. 

134.  Turn  to  Fig.  19,  page  379.     If  the 
radius  of  the  circle  is  8  in.,  what  is  the  area 
of  the  circle  ?    Of  the  hexagon  ?    Of  each 
segment  cut  off  by  the  inscribed  hexagon  ? 

135.  A  circle  10  in.  in  diameter  is  cut  from  a  square  1  ft.  in 
diameter.  What  is  the  area  of  the  remaining  surface  ? 

136.  The  circles   in   Fig.   23  have   the 
same  center.     The  diameter  of  the  smaller 
circle  is  16  in.   and  the  diameter  of  the 
larger  circle  28  in.     Find  the  area  of  each 
circle.     Find  the  area  of  the  circular  ring 
which  is  left  when  the  smaller  circle  is  cut 
from  the  larger. 

137.  Circles    which     have    a    common 
center  are  called  Concentric  Circles. 

Draw  two  concentric  circles,  one  with  a  radius  of  5  inr.,  and 
the  other  with  a  radius  of  7  in.  Find  the  area  of  the  circular 
ring  which  lies  between  their  circumferences. 

138.  If  the  larger  circle  in  Fig.  23 
were  16  in.  in  diameter  and  the  circular 
ring  were  2  in.  wide,  what  would  be  the 

G    diameter  of  the  inner  circle  ?    Its  area  ? 

139.  Draw  a  circle,  and  circumscribe 
a  square  about  it  as  in  Fig.  24.     If  the 
radius  of  the  circle  is  3  in.,  how  long  is 

FIG.  24.  one  side  of  the  square  ? 


FIG.  23. 


382 


MEASUREMENTS   AND   CONSTRUCTIONS 


140.  If  a  side  of  the  circumscribed  square  is  8  ft.,  how 
long  is  the  circumference  of  the  circle?  The  perimeter  of 
the  irregular  figure  FAE  ?  What  is  the  area  of  FAE  ? 

The  area  of  a  circle  may  be  easily  found  by  a  formula  derived  from 

the  formula  A  =  —  •     C  =  —  x  2  r,  or  —  r.     Substituting  this  value  in 
27  7 

the  formula  A  =  -,  we  have  A  =  —  r  x  -,  or  A  -  —  r2.     Thus,  to  find 


of  r2. 


of  25  = 


the  area  of  a  circle  whose  radius  is  5  in.,  we  take 
Ans.  78^  sq.  in. 

141.  Find  by  the  formula  given  in  the  note  the  area  of  a 
circle  whose  radius  is  9  in.     If  in.     12  in.     3i-  in.     I  in. 

T:  Z  o 

142.  Each  member  of  a  geometry  class  numbering  24  pupils 
constructed   a   pasteboard    cylinder    and    a   pasteboard   cone. 
Supposing  the  diameter  of  each  base  of  those  figures  to  be 
7  cm.,  how  many  square  inches  of  pasteboard  were  in  the 
bases  of  the  figures  ? 

143.  A  grass  plot  in  the  form  of  a  semicircle,  whose  straight 
edge  is  15  ft.,  has  within  it  a  round  bed  of  pansies  5  ft.  in 
diameter.     Represent  and  find  the  area  of  the  grass  plot  not 
occupied  by  the  pansies. 

144.  A  garden  28  ft.  square  has  flower 
beds  arranged  as  in  Fig.  25.     Each  semi- 
circle is  12  ft.  in  diameter,  and  the  small 
circle  is  4  ft.  in  diameter.     Find  the  area 
of  the  ground  space  not  within  the  flower- 
beds. 

145.  Turn  to  Ex.  61,  page  337,  and  find 
the  area  of  the  irregular  figure  which  is 
left  when  the  sectors  are  subtracted  from 
the  square. 

146.  Four  circles  whose  centers  are  A, 
B,  C  and  D,  and  whose  radii  are  7  in.,  are 
placed  as  in  Fig.  26.    What  is  the  area  of 

FIG.  26.  the  square  ABCD  ? 


O 


FIG.  25. 


SOLIDS 


383 


147.  Find    the    area 
between  the  circles. 

148. 
which 


of    the    surface    which    is    included 


FIG.  27. 


The  radius  of  the  large  circle 
is  14  in.  equals  the  diameter  of 
each  of  the  small  circles.  Find  the  area 
of  a  small  circle.  Of  the  large  circle.  Of 
the  irregular  figure  ABCDEF. 

Observe  that  ABCDEF  is  one  half  of  the 
space  remaining  when  the  small  circles  are  sub- 
tracted from  the  large  circle. 


SOLIDS 

NOTE  TO  TEACHER.  All  the  solids  treated  here  are  right  solids,  and 
the  bases  of  the  figures  are  regular  polygons.  Pupils 
should  model  these  solids.  The  problems  that  follow 
require  that  models  should  be  used  as  an  objective  basis 
for  work  until  the  pupils  are  able  to  visualize  the  forms 
accurately. 

149.    A  Prism  is  a  solid  whose  bases  are  poly- 
gons and  whose  sides  are  rectangles. 
Quadrangular 

prism.  Mention   some  objects  that  are  in  the  form 

Bases  Squares.        of  a  prism. 

150.  Copy  Fig.  28  on  paper  or  cardboard,  cut  it  out,  and 
fasten  its  parts  together  so  as  to 
make  a  quadrangular  prism. 

151.  If   the  base  of  the  prism 
were  8  in.  square  and  the  altitude 
of  the  prism  10  in.,  what  would  be 
the  area  of  all  the  surfaces  of  the 
prism  ?      How  many   inch    cubes 
would  equal  it  ? 

152.  What   are   the   cubic   con- 
tents of  a  drawer  which  is  18  in. 
square  and  4  in.  deep  ? 


FIG.  28. 


384 


MEASUREMENTS  AND   CONSTRUCTIONS 


153.    Give  the  reason  for  the  following  rule  for  finding  the 
cubic  contents  of  a  prism. 

Multiply  the  area  of  the  base  by  the  altitude. 

The  volumes  of  all  prisms  and  cylinders  are  found  in 
the  same  way. 

154.  Approximately  1J  cu.  ft.  equal  1  bu. 
How  many  bushels  of  wheat  can  be  stored  in  a 
bin  20  ft.  long,  8  ft.  wide,  and  10  ft.  deep  ? 

Find  the  value  of  the  apples  that  fill  a 
,  6  ft.  wide,  44-  ft.  deep,  at  $.75 


lar 
Bases  Triangles. 


155 

bin  8 
a  bu. 


-T  IG. 


156.  From  the  outline  given  in  Fig. 
29  construct  a  triangular  prism. 

157.  If  each  base  edge  of  the  prism 
you  constructed  were  4  in.  and  each 
lateral    edge   were    8    in.,   how   many 
inches  would  there  be  in  all  the  edges 
of  the  prism  ?     Find  the  area  of  its 
lateral  or  side  surface.     Find  the  area 
of  its  entire  surface.     Find  its  volume. 


158.  Find  the  area  of  the  lateral   surface  of   a  triangular 
prism  whose  altitude  is  9  in.  and  each  side  of  whose  base  is 
5  in.     Find  its  entire  surface.     Find  its  volume. 

159.  Find  the  entire  surface  and  the  volume  of  a  triangular 

prism  each  side  of  whose  base  is  6  in.   and 
whose  altitude  is  4.5  in. 


160.  Given  a  pentagonal  prism,  the  perim- 
eter of  whose  base  is  15  in.  and  whose  altitude 
is  8  in.  Find  the  area  of  a  base.  (See  table 
of  ratios,  p.  376.)  Find  the  entire  surface. 
Find  the  volume.  Find  the  sum  of  all  its 
edges. 


Bases  Pentagons. 


SOLIDS 


385 


161.  Construct  a  hexagonal  prism,  the  perimeter  of  whose 
base  is  18  in.  and  altitude  7  in.     Find  its  entire  surface  and 
volume. 

162.  Given  a  hexagonal  prism  the  perimeter  of  a  face  of 
which  is  24  in.  and  altitude  10  in.     Find  the  sum  of  all  its 

Its  entire  surface.     Its  volume. 


163.  Turn  to  Ex.  103,  p.  377.  Suppose  the  square  GHJK 
to  represent  the  base  of  a  quadrangular  prism  of  wood  10  in. 
high.  Suppose  the  wood  to  be  cut  away  until  a  hexagonal 
prism  of  the  same  height  remains,  whose  base  is  represented 
by  the  hexagon  ABCDEF.  Find  the  number  of  cubic  inches 
cut  away. 

164.  Eeproduce   the   square  ABCD. 
With  D  as  a  center,  and  DO,  which  is  \ 
the  diagonal,  as  a  radius,  draw  the  arc 
LOM.    With  A,  B,  and  C  as  centers  and 
with  radii  equal  to  DO,  draw  equal  arcs. 
If  the  side  of  the  square  is  8  in.,  how 
long  is  DO?    DL?    AL?   DK?  AE? 

165.  What  is  the  area  of  each  of  the 
triangles    cut    off    from    the    square  ? 

What  is  the  area  of  the  octagon  remaining  ? 

It  can  be  proved  by  geometry  that  if  the  corners  of  a  square  are  cut  off 
by  the  method  given  in  Ex.  164,  a  regular  octagon  will  be  left.  This 
method  is  used  by  carpenters  in  marking  off  the  end  of  a  square  piece 
of  lumber  in  order  to  change  it  to  an  octagonal  form. 

166.  Find  the  area  of  the  largest  possible  regular  octagon 
that  can  be  cut  from  a  16-in.  square. 

167.  A  piece  of  lumber  4  ft.  long  and  16  in.  square  was 
changed  into  an  octagonal  prism  and  used  as  a  newel  post. 
The  newel  post  was  as  large  as  it  could  be  made  from  the  piece 
of  lumber.     How  many  cubic  inches  of  wood  were  cut  away  ? 
How  many  cubic  inches  were  in  the  newel  post  ? 

HORN.    GRAM.    SCH.    AR. 25 


386 


MEASUREMENTS   AND   CONSTRUCTIONS 


Cylinder. 
Bases  Circles. 


168.  As  the  number  of  sides  of  a  regular 
prism  is  increased,  the  base  approaches  more 
nearly  to  a  circle,  and  the  prism  more  nearly 
to  a  cylinder. 

Draw  a  rectangle  and  construct  two  circles 
whose  circumferences  are  equal  to  a  side  of 
the  rectangle  as  in  Fig.  31.  Cut  out  and 
combine  the  figures  so  that  they  inclose  a 
cylinder. 

169.  If  the  diameter  of 
each    circle    were    14   in. 
and  the  shorter  sides  of 
the  rectangle  were  16  in., 
what  would  be  the  area  of 
the  entire  surface  of  the 
cylinder  ?     If  the  area  of 
the  base  of  a  figure  is  154 
sq.  in.,  how  many  cubic 
inches  of  sand  would  be 
required  to  cover  it  to  the 
depth   of   1   in.?     8  in  ? 
What  would  be  the  vol- 
ume of  this  cylinder  ? 

170.  Find    the    entire 

surface  of  a  cylinder  10  in.  high,  the  diameter  of  whose  base 
is  1  ft.  2  in.     Find  its  volume. 

171.  How  is  the  area  of  the  entire  surface  of  a  cylinder 
found?     How  is  the  volume  of  a  cylinder  found? 

172.  How  many  cubic  feet  in  a  circular  cistern  4  ft.  in  diam- 
eter and  7  ft.  deep  ?     How  many  gallons  will  it  hold  ? 

231  cu.  in.  =  1  gal. 

173.  How  many  square  inches  of  tin  are  there  in  a  dozen 
tin  pails  of  cylindrical  shape,  the  diameter  of  each  being  8  in. 
and  the  height  10  in.  ? 


FIG.  31. 


SOLIDS 


387 


174.  A  cylindrical  tank  10  in.   in  diameter  and  28  in.  in 
height  is  full  of  water.     How  many  gallons  will  remain  in  it 
when  a  pail  of  similar  shape  5  in.  in  diameter  is  filled  from  it  ? 

175.  A  cylinder  7  in.  in  diameter  and  8  in.  in  height,  outside 
measurement,  was  placed  within  another  cylinder  14  in.  in  di- 
ameter and  8  in.  in  height,  inside  measurement.     How  many 
cubic  inches  of  space  were  between  the  two  cylinders  ? 

176.  A  grindstone  28  in.  in  diameter  was  worn  off  until  it 
was  21  in.  in  diameter.     If  the  grindstone  was  4  in.  thick,  how 
many  cubic  inches  were  worn  off  ? 

177.  A  solid  whose  base  is  a  polygon  and 
whose  sides  are  triangles  meeting  at  a  com- 
mon point  is  a  Pyramid. 

Construct  an  equilateral  triangle,  and  with 
each  side  as  a  base  construct  an  isosceles 
triangle  as  in  Fig.  32.  Cut  out  the  figure 
and  bring  the  isosceles  triangles  together  in 
such  a  way  that  their  vertices  meet  in  a  common  point  and 
a  triangular  pyramid  is  formed. 

178.  The  point  where  all  the  faces 
of  a  pyramid  meet  is  called  the  Apex 
of  the  pyramid. 

A  perpendicular  from  the  apex  to 
the  base  meets  the  base  at  its  center. 
That  perpendicular  is  the  Altitude  of 
the  pyramid. 

A  line  from  the  apex  to  the  middle 
point  of  a  side  of  the  base  is  perpen- 
It  is  called  the  Slant  Height  of  the 


Triangular  Pyramid 
Base  a  Triangle. 


FIG.  32. 


dicular  to  that  side, 
pyramid. 

If  a  side  of  the  equilateral  triangle  which  you  have  con- 
structed were  10  in.,  what  would  be  the  area  of  the  base  of  the 
pyramid  ?  If  each  of  the  equal  sides  of  the  isosceles  triangles 
were  13  in.,  what  would  be  the  distance  from  the  vertical 


388  MEASUREMENTS   AND   CONSTRUCTIONS 

angle  of  each  triangle  to  the  middle  point  of  each  side  of  the 
base  ?  What  would  be  the  area  of  the  lateral  surface  of  the 
pyramid  ?  Of  the  entire  surface  ? 

179.  If  a  side  of  the  base  of  the  quadrangu- 
lar pyramid  whose  apex  is  A,  is  10  in.,  how 
long  is  the  distance  BC  from  center  of  the 
base  to  middle  point  of  a  side  ?     If  the  alti- 
tude AB  is  12  in.,  how  long  is  the  slant  height  ? 
What  is  the  area  of  the  lateral  surface  of  the 

Quadrangular  pyr-  .  ,  0 

amid      Base   a    pyramid? 

180.  If  a  side  of  a  base  of  a  quadrangular 
pyramid  were  18  in.,  and  the  slant  height  were  12  in.,  how  long 
would  a  lateral  edge  be  ?     Find  the  sum  of  all  the  edges  of  the 
pyramid.     Find  its  entire  surface. 

181.  Given  a  side  of  the  base  of  a  quadrangular  pyramid  40 
in.,  the  altitude  21  in.,  find  the  slant  height.     Find  the  area  of 
all  the  surfaces  of  the  pyramid. 

182.  Given  a  quadrangular  pyramid  whose  base  is  20  in. 
square,  the  slant  height  26  in.     Find  the  entire  surface  of  the 
pyramid.     Find  the  altitude  of  the  pyramid. 

183.  Given  a  hexagonal  pyramid,  one  side  of  the  base  being 
16  in.,  and  a  lateral  edge  17  in.     Find  the  slant  height  of  the 
pyramid.     Find  its  lateral  surface.     Find  the  sum  of  all  its 


184.  It  can  be  proved  by  geometry  that  the  volume  of  a 
pyramid  equals  -J-  of  the  volume  of  a  prism  having  the  same 
base  and  altitude. 

Find  the  volume  of  the  pyramid  described  in  Ex.  178.     Ex. 
179.     Ex.  180.     Ex.  181.     Ex.  182.     Ex.  183. 

185.  Find  the  contents  of  a  pyramid  whose  base  is  a  trian- 
gle, each  side  of  which  is  8  ft.  and  whose  altitude  is  21  ft. 

186.  If  a  prism  of  wood  whose  base  is  a  square  18  in.  in 
dimensions  and  whose  altitude  is  10  in.,  be  cut  away  until 


SOLIDS  389 

a  pyramid  is  left  having  the  same  base  and  altitude  as 
the  prism,  how  many  cubic  inches  of  wood  must  be  cut 
away  ? 

For  data  for  the  following  problems  see  table  of  ratios,  p.  376. 

Given  a  side  of  a  base  6  in.  and  the  altitude  10  in.,  find  the 
volume  of  : 

187.  A  pentagonal  pyramid. 

188.  An  octagonal  pyramid. 

189.  A  hexagonal  pyramid. 

190.  A  pyramid  whose  base  is  a  decagon. 

191.  The  perimeter  of  the  base  of  a  hexagonal  pyramid  is 
54  in.,  and  a  lateral  edge  is  15  in.     Find  a  side  of  the  base. 
Find  the  distance  from  the  center  of  the  base  to  the  vertex 
of  an  angle  of  the  base.     Find  the  altitude  of  the  pyramid. 
Find  its  volume. 

192.  Given  a  quadrangular  prism  and  a  quadrangular  pyra- 
mid.    The  perimeter  of  the  base  of  each  solid  is  5  ft.  6  in. 
The  altitude  of   the   prism  equals  the  slant   height  of    the 
pyramid,  which  is  1  ft.  8  in.     The  lateral  surface  of  the  prism 
equals  how  many  times  the  lateral  surface  of  the  pyramid  ? 
Which  has  the  greater  altitude,  the  quadrangular  prism  or  the 
quadrangular  pyramid  ? 

193.  At   5^  per  square  foot,  what  would  be  the  cost  of 
painting  the  sides  of  a  steeple  which  is  an  octagonal  pyra- 
mid, each  side  of  the  base  being  8  feet  and  the  slant  height 

being  75  feet? 

194.  The   great  pyramid  of    Gizeh   was 
originally  480  feet  high,  with  a  square  base 
764  feet  on  each  side.      How  many   cubic 
feet  of  masonry  were  there  in  it  ? 

195.  As  the  number  of  sides  of  a  pyra- 
Cone.    Base  a  circle.    mid  ig  increased;  the   base  of   the  pyramid 


390  MEASUREMENTS  AND   CONSTRUCTIONS 

approaches   more  closely  to  a  circle  and  the  pyramid  to  a 
cone. 

Draw  and  cut  out  a  sector,  and  also  a  circle  whose  circum- 
ference equals  the  arc  of  the  sector  as  in  Fig.  33.     With  them 

construct  a  cone. 

196.  What  would  be  the  area 
of  the  convex  or  curved  surface 
of  the  cone  if  the  radius  of  the 
sector  from  which  it  were  made 
were  4  in.  and  the  arc  11  in.  ? 
What  would  be  the  circumfer- 
ence of  the  circle  which  forms 
the  base  of  the  cone  ?  The  diam- 
eter ?  The  area? 

r  ]<;.  oo. 

197.  Would  it  be  possible  to  make  a  cone  by  using  as  a  base 
a  circle  exactly  equal  to  the  circle  from  which  a  sector  is  cut  to 
form  the  curved  part  of  the  cone  ?     Explain. 

198.  What  is  the  area  of  the  curved  surface  of  a  cone  if  the 
circumference  of  the  base  is  4  ft.  8  in.  and  the  slant  height  is 
2  ft.  9  in.  ?     What  is  the  area  of  the  entire  surface  ? 

199.  Can  a  cone  be  constructed  having  the  diameter  of  its 
base  10  in.  and  the  slant  height  4  in.  ?     Explain. 

200.  How  many  square  inches  of  tin  are  needed  to  make  a 
funnel  in  the  shape  of  a  cone,  the  circumference  of  the  base 
being  5  in.  and  the  slant  height  4£  in.  ? 

201.  Four  conical  towers,  each  having  a  diameter  of  4  ft.  and 
a  slant  height  of  12  ft.,  ornament  a  pavilion  in  a  park.     Find 
the  cost  of  gilding  them  at  15  ^  per  square  foot. 

202.  Find  the  entire  surface  of  a  cone  the  radius  of  whose 
base  is  24£  in.  and  the  slant  height  of  which  is  4  ft.  2  in. 

203.  A  line  from  the  apex  of  a  cone  to  the  center  of  its  base 
is  perpendicular  to  the  base.     It  is  the  altitude  of  the  cone. 
What  is  the  slant  height  of  a  cone  whose  altitude  is  24  cm.,  if 


SOLIDS  391 

the  diameter  of  the  base  is  14  cm.  ?     What  is  the  area  of  the 
entire  surface  of  the  cone  ? 

204.  As  a  cone  is  a  pyramid  of  an  infinitely  great  number    ,   ^ 
of  sides,  its  volume  is  equal  to  J  that  of  a  cylinder  whose  alti-  ^ 
tude  and  base  are  respectively  equal  to  those  of  the  cone. 

Find  the  volume  of  a  cone  whose  diameter  is  5  in.  and  alti- 
tude 10  in. 

Find  volumes  of  cones  having  the  following  dimensions : 

205.  Kadius  10  in.,  altitude  2  ft. 

206.  Diameter  15  in.,  altitude  11  in. 

207.  Circumference  5  ft.  6  in.,  altitude  1  ft.  10  in. 

208.  Radius  5  in.,  slant  height  1  ft.  1  in. 

209.  What  is  the  volume  of  the  largest  possible  cone  that 
could  be  cut  from  a  prism  1  ft.  long,  whose  base  is  8  in.  square  ? 

210.  A  cylinder  whose  diameter  is  8  in.  and  altitude  10  in. 
is  cut  entirely  across,  parallel  to  its  base  at  a  distance  of  3  in. 
from  its  base.      What  kind  of  solids  are  formed  ?     What  is 
the  area  of  a  base  of  each  ? 

211.  When  a  solid  is  cut  entirely  through  in  such  a  way 
that  two  plane  surfaces  are  formed,  the  surfaces   are  called 
Sections. 

Represent   a    section    made  by   cutting  across   a  cylinder 
in  such  a  way  that  the  section  is  not  parallel  to  the  base. 

212.  Sections  parallel  to  their   bases  were  made  of  (a)  a 
quadrangular  prism,  (6)  a  hexagonal  pyramid,  (c)  a  cone,  (d)  a 
triangular  pyramid.     What  was  the  shape  of  each  section  ? 

213.  Represent  a  section  not  parallel  to  the  base  of  each  of 
the  above  figures. 


Frustum  of  Pyramid.  Frustum  of  Cone. 


392  MEASUREMENTS  AND  CONSTRUCTIONS 

214.  If  a  pyramid  or  a  cone  is  cut  by  a  plane  parallel  to  its 
base,  the  part  below  the  plane  is  called  a  Frustum  of  the  pyra- 
mid or  of  the  cone.     See  illustrations  on  page  391. 

Construct  a  frustum  of  a  cone  or  of  a  pyramid. 

215.  Each  of  the  bases  of  the  frustum  of  a  triangular  pyra- 
mid is  what  figure  ? 

216.  If  each  side  of  the  lower  base  of  a  triangular  pyramid 
is  10  in.,  each  side  of  the  upper  base  is  8  in.,  and  the  slant 
height  is  7  in.,  what  is  the  area  of  the  lateral  surface  ? 

If  we  find  the  area  of  one  of  the  trapezoids  that  compose  the  lateral 
surface  of  the  frustum  of  a  pyramid  and  then  multiply  that  area  by  the 
number  of  trapezoids  we  shall  have  the  area  of  the  lateral  surface,  but 
it  is  more  convenient  to  find  that  area  by  multiplying  the  average  length 
of  the  perimeters  of  the  upper  and  lower  bases  by  the  slant  height. 

217.  Find  by  each  method  the  area  of  the  lateral  surface  of 
the  frustum  of  a  triangular  pyramid  of  the  following  dimen- 
sions, and  then  try  to  find   why  the  results  are  the  same: 
Edge  of  lower  base  5  in.,  edge  of  upper  base  3  in.,  slant  height 
8  in. 

218.  The  area  of  the  upper  base  of  the  frustum  of  a  square 
pyramid  is  100  sq.  in.,  the  area  of  the  lower  base  144  sq.  in., 
and  the  slant  height.  10  in.     Find  the  entire  surface. 

219.  A  quadrangular  pyramid,  each  side  of  whose  base  is 
16  in.,  is  cut  by  a  plane  so  that  each  side  of  the  upper  base  of 
the  frustum  is  11  in.  long.     The  slant  height  of  the  frustum  is 
10  in.     What  is  the  entire  surface  of  the  frustum  ? 

220.  What  is  the  lateral  surface  of  a  frustum  of  a  hexagonal 
pyramid,  the  perimeter  of  the  lower  base  being  42  in.,  that  of 
the  upper  base  24  in.,  and  the  slant  height  being  5  in.  ? 

221.  Find  the  lateral  surface  of  the  frustum  of  a  hexagonal 
pyramid  if  a  side  of  the  lower  base  is  17  in.,  a  side  of  the  upper 
base  15  in.,  and  the  slant  height  is  1  ft. 


SOLIDS  393 

222.  Find  the  convex  or  curved  surface  of  a  frustum  of  a 
cone,  the  upper  base  of  which  is  40  in.,  the  lower  base  60  in., 
and  the  slant  height  10  in. 

Remember  that  the  frustum  of  a  cone  is  a  frustum  of  a  pyramid  of  an 
infinitely  great  number  of  sides. 

Find  missing  measurements  in  frustums  of  cones : 


Circurn. 
upper  base 

Circum. 
lower  base 

S.  height 

Convex  surface 

223.              8  in. 

1ft. 

Tin. 

9 

224.    2  ft.  3  in. 

3  ft.  2  in. 

8  in. 

9 

225.           11  in. 

15  in. 

? 

78  sq.  in. 

226.    1  ft.  4  in. 

2  ft.  6  in. 

9 

1  sq.  ft.  86  sq.  in. 

227.  How  many  square  feet  of  tin  are  used  in  constructing  a 
tin  pail  in  the  shape  of  a  frustum  of  a  cone  whose  smaller  base 
is  9  in.  in  diameter,  upper  base  1  ft.,  and  whose  slant  height  is 
1  ft.  2  in.,  no  allowance  being  made  for  overlapping  at  the 
seams  ? 

228.  If  a  perfectly  round  ball  7  in. 
in  diameter  were  cut  into  two  hemi- 
spheres, A  and  B,  what  would  be 
the  area  of  each  plane  surface  of  the 
hemispheres  ? 

229.  It  can  be  proved  by  geometry 

that  the  curved  surface  of  a  hemi- 
Hemispheres 

sphere  is  exactly  twice  as  great  as  its 

plane  surface.     What,  then,  would  be  the  area  of  the  outside 
surface  of  the  ball  mentioned  in  Ex.  228  ? 

230.  Find  the  surface  of  a  sphere  whose  diameter  is  1  ft. 
9  in.     1  ft.  51  in. 

231.  Find  the  surface  of  a  sphere  whose  radius  is  8  in. 

232.  A  flagstaff  is  surmounted  by  a  ball  1  ft.  in  diameter. 
Find  the  cost  of  gilding  it  at  30  4  a  square  foot. 


394  MEASUREMENTS   AND   CONSTRUCTION 

233.  What  is  the  entire  curved'  surface  of  a  hemispherical 
dome  whose  height  is  35  ft.  ? 

234.  If    the   earth   were   an   exact   sphere   8000   miles   in 
diameter,  what  would  be  the  area  of  its  surface  ? 

235.  Solids    that    have    the    same    shape    are    said  to   be 
Similar  Solids. 

Think  of  two  boxes  of  the  same  shape,  each  dimension  of 
the  larger  box  being  twice  the  corresponding  dimension  of 
the  smaller.  If  the  larger  box  is  8  in.  by  4  in.  by  2  in.,  what 
are  the  dimensions  of  the  smaller  ?  What  is  the  area  of  the 
surfaces  of  each  ?  What  is  the  volume  of  each  ?  What  is  the 
ratio  of  their  areas  ?  Of  their  volumes  ? 

236.  In  the  case  of  similar  figures,  every  line  of  one  figure 
has  a  corresponding  or  Homologous  Line  on  the  other  figure, 
and  every  angle  on  one  figure  has  a  Homologous  Angle  on  the 
other  figure. 

Take  two  similar  right  triangles  of  different  dimensions  and 
point  out  the  homologous  lines  and  angles. 

SUGGESTION  TO  TEACHER.  Similar  solids  should  be  handled  and  ex- 
amined by  the  pupils.  The  magnitudes  of  their  homologous  angles,  lines, 
surfaces,  and  volumes  should  be  compared  until  the  following  principles 
are  realized. 

1.  On  similar  solids,  homologous  angles  are  equal. 

2.  On  similar  solids,  any  two  homologous  lines  are  to  each  other 
as  any  other  two  homologous  lines. 

3.  On  similar  solids,  homologous  surfaces  are  to  each  other  as 
the  squares  of  their  homologous  lines. 

4.  The  volumes  of  similar  solids  are  to  each  other  as  the  cubes 
of  their  homologous  lines. 

237.  A.  side  of  a  base  of  a  quadrangular  pyramid  is  6  in. 
The  altitude  of  the  pyramid  is  4  in.     What  is  the  slant  height? 
The  sides  of  the  base  of  a  similar  pyramid  are  each  12  in. 
What  is  the  altitude  ?     The  slant  height  ? 


SOLIDS  395 

238.  Find   the   area  of   one  of  the   sides  of    the  smaller 
pyramid.     Of  one  of  the  sides  of  the  larger.     Find  the  ratio 
of  their  areas.     Find  the   volume   of  the   smaller   pyramid. 
Of  the  larger.     Find  the  ratio  of  their  volumes. 

239.  Image   two    similar    quadrangular    pyramids    whose 
homologous  lines  are  in  the  ratio  of  1  to  3.     What  is  the  ratio 
of  the  areas  of  their  bases  ?     If  a  lateral  edge  of  the  smaller 
pyramid   is  8  in.,  what  is   the  lateral  edge  of  the  larger? 
Assume  dimensions  for  the  figures,  and  find  the  areas  of  a 
triangular  side   of  each   solid,  and   also  the   ratio   of  those 
areas. 

240.  The  slant  height  of  the  frustum  of  a  hexagonal  pyramid 
is  12  in.     A  side  of  the  lower  base  is  8  in.     A  side  of  an  upper 
base  is  6  in.     Find  the  perimeters  of  the  bases  of  a  similar 
frustum,  the  slant  height  of  which  is  3  in. 

241.  Find  the  lateral  surface  of  each  of  those  frustums. 
Find  the  ratio  of  the  surfaces. 

242.  If  a  bucket  8  in.  in  diameter  holds  3  gal.,  how  many 
gallons  can  be  poured  into  a  bucket  of  similar  shape  whose 
diameter  is  16  in.  ? 

243.  If  it  takes  110  sq.  in.  of  tin  to  make  a  milk  can  5  in. 
in  diameter,  how  many  square  feet  of  tin  will  be  required  to 
make  a  similar  can  20  in.  in  diameter  ? 

244.  If  the  volume  of  a  cone  whose  altitude  is   6  in.   is 
54  cu.  in.,  how  many  cubic  inches  are  there  in  a  similar  cone 
whose  altitude  is  10  in.  ? 

245.  A  solid  iron  ball  weighs  7  Ib.     What  would  be  the 
weight  of  a  similar  ball  whose  diameter  is  twice  as  great  as 
that  of  the  first  ? 

246.  The  cost  of  gilding  a  vase  was  $  1.60.     What  would 
be  the  cost  of  gilding  a  larger  vase  of  the  same  shape,  the 


396  MEASUREMENTS  AND   CONSTRUCTIONS 

larger  vase  being  twice  as  high  as  the  smaller  ?  If  the  cubic 
contents  of  the  larger  vase  were  328  cu.  in.,  what  were  the 
cubic  contents  of  the  smaller  ? 

247.  Two  similar  cylinders  are  respectively  2  in.  and  8  in. 
in  diameter.     If -a  section  is  made  parallel  to  the  base  of  each, 
what  is  the  ratio  of  the  area  of  the  section  of  the  greater 
cylinder  to  that  of  the  less  ? 

248.  A  pyramid  whose  base  is  6  ft.  square  and  whose  alti- 
tude is  4  ft.  is  cut  by  a  plane  parallel  to  its  base  and  2  ft.  above 
it.     The  pyramid  above  the  cut  equals  what  part  of  the  origi- 
nal pyramid  ? 

249.  If  a  coal  bin  4  ft.  long  holds  20  bu.  of  coal,  how  many 
bushels  can  be  put  into  a  bin  of  similar  shape  which  is  8  ft. 
long? 

250.  If   a  city  lot  one  side   of  which  is  80  ft.    is  worth 
$  12,000,  what  is  the  value  of  a  lot  of  the  same  shape,  the 
corresponding  side  of  which  is  40  ft.  ? 

251.  16  sq.  ft.  of  galvanized  iron  were  used  in  making  a 
water  tank  for  a  stove.     How  many  square  feet  must  be  used 
to  make  a  tank  of  the  same  shape,  each  edge  of  which  is 
twice  that  of  the  first  tank  ?     If  the  first  tank  held  30  gal., 
how  many  gallons  would  the  second  tank  hold  ? 

252.  A  certain  freight  car  contains  24,000  cu.  ft.  of  space. 
How  many  cubic  feet  of  space  will  be  contained  in  a  model  of 
this  car,  which  is  -^  as  long  as  "the  original  ? 

253.  What  is  the  ratio  of  a  diagonal  of  a  face  of  a  liter  to 
a  diagonal  of  a  face  of  a  stere  ? 

254.  What  is  the  ratio  of  the  sum  of  the  surfaces  of  a  liter 
to  the  sum  of  the  surfaces  of  a  stere  ?     What  is  the  ratio  of 
a  liter  to  a  stere  ? 

255.  If  each  of  the  sides  of  a  polygon  is  trebled,  the  result- 
ing polygon  equals  how  many  times  the  original  polygon  ? 


ARCS   AND   ANGLES  397 

256.  If  each  of  the  lines  of  a  frustum  of  a  hexagonal  pyra- 
mid were  made  5  times  as  long,  the  volume  of  the  resulting 
solid  would  be  how  many  times  the  original  solid  ? 

257.  If  a  straw  stack  5  ft.  high  contains  3  tons,  how  many 
tons  are  there  in  a  stack  of  similar  shape  10  ft.  high  ? 

258.  How  many  balls  of  lead  2  in.  in  diameter  will  weigh 
as  much  as  a  ball  of  lead  8  in.  in  diameter  ?     What  is  the 
ratio  of  the  sum  of  all  the  surfaces  of  the  2-inch  balls  to  the 
surface  of  the  8-inch  ball  ? 

ARCS  AND  ANGLES 

259.  Circumferences  of  circles  are  considered  to  be  divided 
into  360  equal  parts  called   degrees,  marked  °.     How  many 
degrees  in  a  semicircumference ?     In  a  quadrant? 

260.  Over  how  many  degrees  does  the  minute  hand  of  a 
clock  pass  in  15  min.  ?    In  45  min.  ? 

261.  How  many  degrees  are  described  by  the  hour  hand  of 
a  clock  in  4  hr.  ?     In  2£  hr.  ?     In  5  hr.  30  min.  ? 

262.  If  a  circumference  is  divided  by  a  chord  in  such  a 
way  that  the  greater  arc  is  4  times  the  less,  how  many  degrees 
are  there  in  each  arc  ? 

263.  If  a  regular  hexagon  is  inscribed  in  a  circle,  how  many 
degrees  are  there  in  each  arc  cut  off  by  a  side  of  the  hexagon  ? 

264.  How  many  degrees  are  there  in  each  arc  subtended  by 
a  side  of  a  regular  inscribed  octagon  ?     By  a  side  of  a  regular 
inscribed  decagon  ?     Dodecagon  ?    Heptagon  ?    Nonagon  ? 

265.  An  arc  20  ft.  long  equals  how  many  degrees  of  a  cir- 
cumference 160  ft.  long  ?     Of  a  circumference  240  ft.  long  ? 

266.  A  horse  trotted  1760  ft.  on  a  circular  race  track  1  mi. 
in  length.     Over  how  many  degrees  of  its  circumference  did 
he  pass  ? 

267.  If  a  circumference  is  24  in.,  how  long  is  an  arc  of  60°  ? 
90°  ?    30°  ?    150°  ?    120°  ?    45°  ?    75°  ?    108°  ? 


898  MEASUREMENTS  AND  CONSTRUCTIONS 

268.  Find  the  length  of  an  arc  of  110°  of  a  circumference 
which  is  20  ft.     Of  a  circumference  whose  radius  is  7  ft. 

269.  How  long  is  an  arc  of  45°  in  a  circle  whose  diameter 
is  6  ft.  5  in.  ?     In  a  circle  whose  radius  is  4  ft.  8  in.  ? 

270.  How  long  is  an  arc  subtended  by  the  side  of  a  regular 
pentagon  inscribed  in  a  circle  whose  circumference  is  40  ft.  ? 
In  a  circle  whose  diameter  is  10  ft.  ? 

271.  In  order  to  make  calculations  more  exact,  a  degree  is 
divided  into  60  equal  parts  called  minutes,  marked  ',  and  a 
minute  is  divided  into  60  equal  parts  called  seconds,  marked  ". 

TABLE  OF  ANGULAR  MEASURE 
60  seconds  (")  =  1  minute  (') 
60  minutes      =  1  degree  (°) 

Do  not  confound  minutes  and  seconds  that  are  measures  of  arcs  and 
angles  with  minutes  and  seconds  that  are  measures  of  time. 

An  arc  which  is  one  minute  is  what  part  of  a  circumference  ? 
An  arc  which  is  one  second  is  what  part  of  a  circumference  ? 

272.  How  many  minutes  in  25°  30'  ? 

273.  How  many  minutes  in  4£°  ?     In  720"  ?     In  5°  55' ? 

274.  How  many  seconds  in  35' 25"  ?     In  8°  10' 20"?     In 
90°  15'  25"  ? 

275.  Express  50°  15'  30"  as  seconds.     As  minutes.     As  de- 
grees. 

Express  in  each  denomination  of  angular  measure : 

276.  7°  10'  20".  281.    9'  20". 

277.  8°  10".  282.    10°  45". 

278.  4' 30".  283.    8°  6' 6". 

279.  7°  20' 40".  284.    4' 50". 

280.  15°  15'  15".  285.    12°  12'  12". 

286.   Express  in  degrees,  minutes,  and  seconds,  42784  sec. 
31125  sec.     57241  sec. 


ARCS   AND  ANGLES  399 

287.    Express  in  higher  denominations,  97860  sec.    77825  sec. 


Add: 
288.    8°    5'  50" 
6°  11'  27" 

289.     6°    7'  24" 
20°  37'  48" 

290.     8°  29'  33" 
17°  31'  47" 

Find  difference  : 
291.    8°  7'  25" 
3°  8'  16" 

292.  24°  16'  38" 
7°  19'  49" 

293.  85°  21'  36" 

17°  27'  54" 

Multiply  : 
294.    41°  17'  25" 
8 

295.  16°  17'  19" 
9 

296.  23°  28'  39" 
11 

Divide : 

297.  15)18°  36'  45"  299.    24)13°  19'  28" 

298.  11)19°  36'  48"  300.    15)7°  8'  43" 

301.   The  sum  of  two  arcs,  one  49°  1'  28",  the  other  16°  38'  59", 
is  how  much  less  than  the  whole  circumference  ? 

B  302.    How  many  degrees,  minutes,  and  seconds  in  $ 

of  a  circumference  ? 

303.    Think  of  the  position  of  the  hands  of  a  clock 
'°  34    at  12  o'clock.     Imagine  two  equal  lines  BO  and  AO 
in  the  same  position  (Fig.  34).    Suppose  AO  to  remain 
fixed,  and  that  BO  makes  a  complete  revolution  around  the 
point  0  and  returns  to  its  former  position,  the  point  B  de- 
scribing a  circle.     At  the  different  stages  of  its  revolution  BO 
makes  different  angles' with  AO,  corresponding  in  the  number 
of  degrees  to  the  arcs  described.     As  the  angle  of  the 
whole  revolution  is  considered  an  angle  of  360°,  the 
angle  formed  by  the  two  lines,  when  \  of  the  revo- 
lution has  been  made,  equals  90°,  or  a  right  angle. 
When  BO  has  made  £  of  a  revolution,  and  the  two 
lines  are  so  placed  that  each  is  a  continuation  of  the 
other  (Fig.  35),  the  angle  formed  by  them  is  an  angle 
of  180°,  or  a  straight  angle. 

How  many  right  angles  form  a  straight  angle  ? 


400  MEASUREMENTS   AND   CONSTRUCTIONS 

304.  How  many  degrees  are  there  in  the  sum  of 
the  angles  a  and  b  ? 

305.  -Reproduce  Fig.  36  several  times,  changing  the 
dividing  line  between  the  angles  to  make  it  lie  in  dif- 
ferent directions.     The  sum  of  the  angles  is  always 
how  many  degrees  ? 

306.  Two  angles,  whose  sum  is  equal  to  180°,  are 
said  to  be  Supplements  of  each  other. 

How  many  degrees  are  there  in  the  supplement  of  an  angle 
of  100°  ?     Of  179£°  ?     Of  3°  ?     Of  a  right  angle  ? 

307.  How  many  degrees  are  there  in  the  supplement  of  an 
angle  which  is  |  of  a  right  angle  ? 

308.  How  much  greater  or  less  than  its  supplement  is  an 
angle  of  80°?     90°?     130°?     170°?     500°?     75°? 

309.  How  many  degrees  are  there  in  an  angle  whose  sup- 
plement is  twice  the  given  angle  ? 

310.  How  many  degrees  are  there  in  an  angle  which  is  twice 
its  supplement? 

311.  What  is  the  ratio  of  an  angle  of  110°  to  its  supple- 
ment ?     To  a  straight  angle  ?     To  a  right  angle  ? 

312.  A  fan,  which  when  opened  was  semicircular  in  shape, 
was  opened  -|  of  its  extent.     What  angle  was  formed  by  the 
outside  edges  of  the  sticks  ? 

313.  A  branch  of  a  tree   made  an  angle  of  45°  with  the 
trunk  of  the  tree.    What  was  the  supplement  of  that  angle  ? 

314.  How  many  degrees  are  there  in  the  angle  formed  by 
the  hands,  of  a  clock  at  2  o'clock  ?     4  o'clock  ?     6  o'clock  ? 

315.  How  many  degrees  are  there  in  the  supplement  of  an 
angle  formed  by  the  hands  of  a  clock  at  1.30  P.M.  ? 

316.  What  kind  of  an  angle  is  the  supplement  of  an  acute 
angle  ?     Of  a  right  angle  ?     Of  an  obtuse  angle  ?     Explain. 


ATCCS  AND   ANGLES 


401 


FIG.  37. 


317.    When  is  an  angle  greater  than  its  supplement?     Less 
than  its  supplement? 

318.  How  many  degrees  are  there  in 
the   sum   of   all   the   angles,  a,   6,  c,  d, 
formed  at  a  given  point  and  011  the  same 
side  of  a  straight  line  ?     Explain. 

319.  An  angle  formed  by  radii  of  a 
circle  contains  just  as  many  degrees  as 
the  arc  which  is  included  between  the 
ends  of  the  radii. 

0  is  the  center  of  the  circle  of  which 
the  arc  AB  is  60°.  How  many  degrees 
are  there  in  the  angle  AOB  ?  In  BOG  ? 
In  the  straight  angle  AOC? 

In  naming  an  angle  by  three  letters,  the  let- 
ter at  the  vertex  is  placed  between  the  other  two 
letters. 

320.  The  angle  ABC  is  a  right  angle. 
ABD  is  10°  more  than  DBG.     QBE  is  4 

E 

times    EBF.      How    many    degrees    in 
"F  DBG?    In  ABD?    In  EBF?    In  EBC? 
InDBE?     In  ABE? 

321.  Three  angles,  a,  6,  and  c,  are  formed  at  the  same  point 
and  on  the  same  side  of  a  straight  line,     a  =  3  times  b}  and 
c  =  4  times  b.     How  many  degrees  are  there  in  each  ? 

Let  x  =  number  of  degrees  in  angle  b. 

322.  Four  angles,  a,  b,  c,  and  d,  are  formed  on  one  side  of  a 
straight  line  at  the  same  point,     b  has  10°  more  than  a,  c  has 
10°  more  than  &,  and  d  has  10°  more  than  c.     How  many  •  de- 
grees are  there  in  each  ? 

323.  How  many  degrees  are  there  in  the  sum  of  an  angle 
of  17°  3'  15"  and  an  angle  of  17°  4'  21"? 

324.  How  many  degrees,  minutes,  and  seconds  are  there  in 
the  supplement  of  an  angle  of  105°  15'  20"  ? 

HORN.    GRAM.    SCH.    AR.  — 26 


B 

FIG.  39. 


402 


MEASUREMENTS   AND   CONSTRUCTIONS 


325.  How  many  degrees,  minutes,  and  seconds  are  in  the  sup- 
plement of  an  angle  which  is  4  times  an  angle  of  7°  40'  50"  ? 

326.  Give  the  measurement  of  an  angle  which  is  5  times 
the  supplement  of  an  angle  of  150°  10'  24". 

327.  Give  the  measurement  of  an  angle  whose  supplement 
is  6  times  an  angle  of  10°  20'  30". 

328.  The  figure  below  represents  a  Protractor,  a  device  for 
measuring  or  constructing  angles.     The  point  at  the  center 
marked  A  in  this  figure  must  always  be  placed  at  the  vertex 
of   the   angle   that  is  measured  or  constructed.     The  degrees 
corresponding  to  the  angle  are  marked  upon  the  arc. 

Make  a  protractor.     Lay  off  by  means  of  it  an  angle  of  20°. 


329.  Draw  an  angle  and  find  by  the  protractor  the  number 
of  its  degrees. 

330.  Find  by  the  protractor  how  many  degrees  there  are  in 
an  angle  of  an  equilateral  triangle.     Of  a  regular  hexagon. 

SUGGESTION  TO  TEACHER.  Let  pupils  measure  angles  found  in  decora- 
tive designs,  as  in  wall  paper,  carpet,  parquetry,  also  the  angles  formed  by 
branching  stems,  the  angles  of  crystals,  and  of  other  natural  objects. 

331.  How  many  degrees  are  there  in  each  angle  of  a  square  ? 
In  all  the  angles  of  a  square  ? 


ARCS   AND   ANGLES 


403 


332.  Draw  a  square.  With  each  corner  of  the  square  as  a 
center  and  with  a  radius  less  than  ^  the  side  of  the  square  draw 
arcs  inside  the  square  ending  in  its  sides. 
(See  p.  337,  Fig.  10.)  Cut  out  the  4  sectors 
and  place  them  so  that  their  vertices  are 
at  the  same  point.  What  figure  is  formed  ? 

333.  Plow  many  degrees  are  there  in 
all  the  angles  a,  &,  c,  dt  e  formed  about  a 
common  point  in  Fig.  40  ? 

334.  Angle  1  (Fig.  41)  is  89°  10',  angle 
2  is  48°  30',  angle  3  is  90b  5'.     How  many 
degrees  in  angle  4  ? 

335.  How  many  degrees  are  there  in 
each  of  8  equal  angles  whose  vertices  are 

Fl(,  41  .afr-the  same  point  ? 

336.  There  are  4  angles  a,  6,  c,  and  d  around  a  common  point, 
a  =  3  times  b,  c  =  twice  6,  and  d  =  twice  a.     Find  the  number 
of  degrees  in  each.     Represent. 

337.  Construct  around  a  common  point  an  angle  of  75°,  an 
adjacent   angle  of   65°,   and   another   adjacent   angle   of  80°. 
How  many  degrees  are  there  in  the  remaining  angle  ? 

338.  How  long  is  the  perimeter  of  a  sector  of  60°  of  a  circle 
whose  radius  is  15  in.  ?     What  is  the  area  of  the  sector  ? 

339.  If  a  sector  of  45°  were  cut  from  a  circle  whose  diameter 
is  20  in.,  how  long  would  be  the  perimeter  of  the  remaining 
figure  ?     What  would  be  its  area  ? 

340.  Find  the  perimeter  and  the  area  of  a  sector  of  40°  of 
a  circle  whose  radius  is  18  in. 

341.  What  is  the  area  of  the  figure  that  remains  when  a 
sector  of  36°  is  cut  from  a  circle  whose  radius  is  3  ft.  6  in.  ? 
How  long  is  the  perimeter  of  the  figure  ? 


404 


MEASUREMENTS  AND   CONSTRUCTIONS 


342.  Construct  an  angle  of  50°,  an  angle  of  60°,  and  an 
angle  of  70°.  Cut  them  out  and  place  them  side  by  side  so  that 
their  vertices  are  at  a  common  point.  What  kind  of  an  angle 
do  they  form  ? 


FIG.  42. 


343 .  Draw  and  cut  out  a  triangle,  mark- 
ing its  angles  1,  2,  3,  as  in  Fig.  42. 

Cut  off  the  corners  1  and  3  as  in 
Fig.  43.  Place  them  beside  corner  2  as 
in  Fig.  44. 

The  angles  will  have  their  vertices  at 
a  common  point  and  will  cover  all  the 
surface  around  that  point  on  the  same 
side  of  a  straight  line.  Hence  they  are 
equal  to  two  right  angles. 


FIG.  44. 


344.  Kepeat  the  experiment  with  triangles  of  different 
shapes  until  you  see  the  truth  of  the  following: 

The  sum  of  the  angles  of  a  triangle  is  equal  to  two  right  angles, 
or  180°. 

By  geometry  this  is  proved  to  be  true  in  all  cases. 

345.  How    many    degrees     are 
there  in  angle  A  of  Fig.  45? 

346.  If  A  were  83°  and  B  were 
75°,  how  many  degrees  would  angle 
C  be? 

347.  If  A  were  91°  and  C  41°, 
how  many  degrees  would  angle  B 
be? 

348.  If  one  angle  of  a  triangle  is  a  right  angle,  how  many 
degrees  are  there  in  the  sum  of  the  other  two  angles  ?  What 
kind  of  angles  are  they  ? 


FIG.  45. 


ARCS  AND  ANGLES  405 

349.  If  one  of  the  angles  of  a  right  triangle  is  38°,  how  many 
degrees  are  there  in  the  other  acute  angle  ? 

350.  At  one  extremity  of  a  line  construct  an  angle  of  35°. 
Construct  an  angle  of  75°  at  the  other  end  of  the  line.     Prolong 
the  added  lines  until  they  meet.    How  many  degrees  are  there 
in  the  angle  formed  by  their  meeting  ? 

351.  Fold  an  isosceles  triangle  so  that  the  equal  sides  coin- 
cide.    Can  you  see  that  the  following  statement  is  true  ? 

In  an  isosceles  triangle  the  angles  opposite  the  equal  sides  are 
equal. 

352.  How  many  degrees  are  there  in  each  angle 
of  the  isosceles  triangle  ABC  ?     Explain. 

353.  How  many  degrees  are  there  in  each  angle 
of  an  isosceles  triangle  in  which  a  base  angle  is 
80°  ?     Explain. 

354.  Draw  two  equal  lines  forming  a  right  angle  and  join 
their  extremities.  What  kind  of  a  triangle  is 
formed?  How  many  degrees  are  there  in 
each  angle? 

355.  Find  the  measurement  of  each  angle 
of  an  isosceles  triangle  in  which  a  base  angle 
is  3  times  the  vertical  angle.     Represent. 

356.  How   many   degrees   are    there    in 
angle  c  ?     How  many  degrees  in  each  angle 
of  the  isosceles  triangle  in  Fig.  47  ? 

357.  An  angle  formed  by  a  side  of  a  polygon  and  a  pro- 
longation of  an  adjacent  side  is  called  an  Exterior  Angle. 

Show  two  exterior   angles   in  Fig.  47,  and  tell  how  many 
degrees  in  each. 

358.  Reproduce  Fig.  47  and  make  an  angle  exterior  to  b. 
How  many  degrees  are  there  in  it  ?     Can  an  isosceles  triangle 
have  two  exterior  angles  at  the  base  unequal  ?    Explain. 


406  MEASUREMENTS  AND   CONSTRUCTIONS 

359.  How  many  degrees  are  there  in  each  angle 
of  the  isosceles  triangle  ABC  (Fig.  48),  whose  base 
is  BC? 

360.  The  triangle  ADO  (Fig.  49)  is  isosceles. 
A,  the  vertical  angle,  is  40°.     CB  is  perpendicular 
to  AD.     How  many  degrees  are  there  in  each  angle 
of  the  triangles  ABC  and  DBC? 

361.  The  triangle  ABC  (Fig.  50)  is  isosceles. 
AE  bisects   the   angle   A.     Angle  DBC  is  100°. 
Find  each  angle  of  the  triangles  AEB  and  AEC. 

362.  In  a  scalene  triangle  ABC,  A  is  a  right 
angle,  and  the  angle  B  is  8  times  the  angle  C. 
How  many  degrees  are  there  in  each  angle  of  the 
triangle  ?     Represent. 

363.  How  many  degrees  apart  are  the  equator 
and  the  north  pole  ?     How  many  degrees  apart 
are   two   places   which   are   on   exactly   opposite 
points  of  the  equator  ? 

Recall  your  knowledge   of  meridians  and  parallels  of 
latitude  by  reference  to  your  geography,  if  necessary. 

364.  How  many  degrees  apart  are  two  places, 
one  of  which  is  on  the  equator,  and  the  other  half 

Fia.  50.        way  between  the  equator  and  the  south  pole  ? 

365.  How  many  degrees  apart  are  two  points,  one  of  which 
is  at  the  south  pole,  and  the  other  30°  from  the  north  pole  ? 

366.  Considering  the  circumference  of  the  earth  as  25,000 
miles,  how  far  apart  are  two  places  on  the  equator  which  are 
60°  apart  ?     180°  apart  ?     120°  apart  ? 

In  the  following  problems  the  equator  and  meridian  circles  are  assumed 
to  be  circles  25,000  miles  in  circumference. 

367.  How  long  is  an  arc  of  the  equator  which  is  50°  ?   70  °  ? 

368.  How  long  is  an  arc  of  50°  of  a  circle  of  latitude  which 
is  1800  miles  in  circumference  ? 


ARCS   AND   ANGLES  407 

369.  An  arc  of  75°  of  a  parallel  of  latitude  120  miles  in  cir- 
cumference equals  how  many  miles  ? 

370.  How  many  miles  from  the  equator  is  a  place  20°  north 
of  it  ?     About  how  many  miles  from  the  equator  are  you  ? 

371.  If  a  ship  sailed  2000  miles  on  the  equator,  through  how 
many  degrees  of  longitude  would  it  pass  ? 

372.  If  a  ship  sailed  200  miles  on  a  circle  of  latitude,  the 
circumference  of  which  was  4000   miles,  through  how  many 
degrees  would  it  pass  ? 

373.  How  many  degrees  from  the  equator  is  a  place  that 
is  500  miles  north  of  it  ? 

374.  A   village   is   2°  10'  30"  east  of  a  certain  city,  and 
another  village  is  17°  49'  30"  west  of  the  city.     How  many 
degrees  apart  are  they  ?     If  the  circle  of  latitude  upon  which 
they  are  situated  is  1200  miles  in  circumference,  how  many 
miles  apart  are  the  villages  ? 

375.  What  is  the  difference  in  longitude  between  a  place 
20°  13'  48"  east  longitude,  and  a  place  15°  15'  55"  west  longi- 
tude ?     Draw  a  diagram  to  illustrate  the  problem. 

Find  the  difference  of  longitude  between : 

376.  A,  17°  15'  30"  east,  and  B,  16°  16'  58"  west. 

377.  C,   40°  30'  20"  east,  and  D,  50°  10'  25"  west. 

378.  E,  17°  13'  21"  west,  and  F,  19°  18'  24"  west. 

379.  G,  41°  16'  29"  east,  and  H,  31°  17'  27"  east. 

380.  J,   61°  16'  38"  east,  and  K,  15°  15'  45"  west. 

381.  L,  6' 17"  east,  and  M,  20'  30"          west. 

382.  A  certain  lighthouse  is  6°  15'  20"  north  of  the  equator, 
and  another  lighthouse  on  the  same  meridian  is  11°  44'  40" 
south  of  the  equator.     How  many  miles  apart  are  they  ? 


408  MEASUREMENTS  AND  CONSTRUCTIONS 

383.  The  earth,  revolves  on  its  axis  once  in  24  hr.    Through 
how  many  degrees  does  a  point  on  the  earth's  surface  turn  in 
one  hour  ? 

384.  A  point  on  the  equator  revolves  at  the  rate  of  how 
many  miles  in  one  hour? 

LONGITUDE   AND  TIME 

As  the  earth's  motion  in  revolving  on  its  axis  is  from  west  to  east, 
when  it  is  sunrise  at  a  certain  place,  points  west  of  that  place  are  still  in 
darkness. 

385.  Suppose  the  sun  to  rise  at  6  o'clock  at  a  certain  place, 
how  much  less  than  6  o'clock  is  the  time  of  day  at  a  place  15° 
west  of  that  place  ?     What  is  the  time  15°  east  of  that  place  ? 

386.  When  it  is  noon  at  Denver,  is  it  A.M.  or  P.M.  at  Bos- 
ton ?    New  York  ?    San  Francisco  ?    Philadelphia  ?   Portland, 
Oregon  ?     Portland,  Me.  ?     Washington,  D.C.  ?     St.  Louis  ? 

387.  Find  the  difference  in  longitude  and  the  direction  from 
Chicago  of  a  place  whose  real  time  is  2  hr.  later  than  Chicago 
time.     3  hr.  earlier.     5J  hr.  earlier.     4  hr.  30  min.  later  ? 

388.  When  it  is  noon  at  Buffalo,  what  time  is  it  at  a  place 
15°  east  of  Buffalo  ?     15°  west  ?    60°  east  ?    40°  west  ? 

389.  When  it  is  midnight  at  St.  Paul,  what  time  is  it  at  a 
place  30°  north  of  St.  Paul  ?    30°  south  ?   30°  west  ?    30°  east  ? 

390.  If  a  point  moves  15°  in  1  hr.,  or  in  60  minutes  of  time, 
how  far  will  it  move  in  one  minute  ? 

391.  If  a  point  moves  \  of  a  degree,  or  15'  in  one  minute  of 
time,  how  far  will  it  move  in  one  second  of  time  ? 

15°  of  longitude  cause  a  difference  of  1  hr.  of  time. 
15'  "  "  "  1  min.  of  time. 

15"  «  "  "  1  sec.  of  time. 

392.  What  is  the  difference  in  longitude  of  two  places  be- 
tween which  there  is  a  difference  of  2  hr.  10  min.  17  sec.  of  time  ? 


LONGITUDE   AND  TIME  409 

Since  a  difference  of  1  hr.  of  time  is  caused  by  a 

hr.  min.  sec.      difference  of  15°  in  longitude,  1  min.  of  time  by  15'  of 

2     10     17      longitude,  and  1  sec.  of  time  by  15"  of  longitude,  we 

15      multiply  the  number  of  hours,  minutes,  and  seconds  by 

40     4       5      15  to  find  the  corresponding  number  of  degrees,  minutes, 

and  seconds  of  longitude. 

393.  When  it  is  4  P.M.  at  Anda  it  is  6.30  P.M.  at  Roseville. 
What  is  the  difference  in  longitude,  and  which  place  is  further 
east? 

Find  difference  in  longitude  and  relative  position  of  places 
whose  simultaneous  time  reckonings  are  as  follows : 

394.  A,  3  hr.  30  min.  P.M.  B,  5  hr.  10  min.  30  sec.  P.M. 

395.  C,  11  hr.  30  min.  A.M.  D,  12  hr.  30  min.  P.M. 

396.  E,  10  hr.  30  min.  30  sec.  A.M.    F,  2  hr.  30  min.  45  sec.  P.M. 

397.  G,  Noon  H,  10  hr.  30  min.  25  sec.  A.M. 

398.  J,  9  hr.  30  min.  P.M.          K,  Midnight. 

399.  L,  8  hr.  40  min.  30  sec.  P.M.  M,  5  hr.  20  min.  10  sec.  P.  M. 

400.  N,  7  hr.  30  min.  15  sec.  A.M.  0, 1  hr.  10  min.  15  sec.  P.M. 

401.  P,  9  hr.  20  min.  10  sec.  A.M.  Q,  3  hr.  5  min.  30  sec.  P.M. 

402.  The  longitude  of  the  following  places  is  reckoned  from 
the  meridian  of  Greenwich. 


Portland,  Me., 

70°  15'  18"  W. 

Chicago,  111., 

87°  37'    0"  W. 

Boston,  Mass., 

71°    3'  50"  W. 

New  Orleans,  La.  , 

90°    5'    0"  W. 

New  York  City, 

74°    0'  36"  W. 

Omaha,  Neb., 

95°  56'  14"  W. 

Pittsburg,  Pa., 

80°    2'    0"  W. 

Paris,  France, 

2°  20'    0"  E. 

Savannah,  Ga., 

81°    5'  26"  W. 

Rome,  Italy, 

12°  28'    0"  E. 

Louisville,  Ky., 

85°  30'    0"  W. 

Vienna,  Austria, 

16°  23'    0"  E. 

Nashville,  Tenn.,  86°  49'    0"  W.      St.  Petersburg,  Russia,  30°  18'    0"  E. 
The  difference  in  time  between  Portland,  Me.,  and  a  place 
west  of  it  is  2  hr.     What  is  the  longitude  of  that  place  ? 

403.   In  what  longitude  is  a  place  that  has  2  hr.  10  min. 
later  time  than  Louisville  ? 


410  MEASUREMENTS  AND   CONSTRUCTIONS 

404.  In  what  longitude  is  a  place  where  it  is  half  past  ten 
in  the  morning  at  the  same  instant  when  it  is  noon  at  Pitts- 
burg? 

What  is  the  longitude  of  a  place  which  has  : 

405.  3  hr.  30  min.  later  time  than  Boston? 

406.  2  hr.  20  min.  earlier  time  than  New  York  City  ? 

407.  1  hr.  10  min.  earlier  time  than  Nashville  ? 

408.  2  hr.  15  min.  later  time  than  New  Orleans  ?   Louisville  ? 

409.  3  hr.  15  min.  30  sec.  later  time  than  New  Orleans  ? 

410.  1  hr.  20  min.  later  time  than  Paris? 

411.  1  hr.  40  min.  later  time  than  Kome  ? 

412.  3  hr.  earlier  time  than  Kome? 

413.  4  hr.  earlier  time  than  St.  Petersburg? 

414.  3  hr.  later  time  than  Washington,  D.C.  ? 

i 

415.  4  hr.  30  min.  later  time  than  Boston  ? 

416.  A  difference  of  75°  45' 15"  between  two  places  causes 
how  much  difference  in  time  ? 

15)  75°  45'  15"          Since  15°  of  longitude  cause  a  difference  of  1  hr.  in 
K      o     T~      time,  15'  of  longitude,  1  min.  in  time,  and  15"  of  lon- 
gitude 1  sec.  in  time,  we  divide  the  number  of  degrees, 
minutes,  and  seconds  of  longitude  by  15  to  find  the  corresponding  num- 
bers of  hours,  minutes,  and  seconds  of  time. 

417.  A  city  is  30°  15'  45"  west  of  a  certain  meridian.     What 
is  the  difference  in  time  between  that  city  and  all  places  on 
that  meridian  ? 

418.  There  are  two  meridians  24°  48'  30"  apart.     When  it  is 
2  P.M.  at  places  on  the  western  meridian,  what  time  is  it  at 
the  places  on  the  eastern  meridian  ? 

What  is  the  difference  in  time  between  the  following  places  : 

419.  Portland,  Me.,  and  Omaha  ? 

420.  Boston  and  Chicago  ? 


LONGITUDE   AND   TIME  411 

421.  Washington,  B.C.,  and  New  Orleans  ? 

422.  Pittsburg  and  Omaha  ? 

423.  New  York  and  Paris  ? 

424.  Boston  and  St.  Petersburg? 

425.  Washington  and  Rome  ? 

426.  Nashville  and  Vienna  ? 

427.  Two  persons,  one  in  Paris  and  the  other  in  Rome, 
agreed  to  read  a  certain  poem  at  the  same  time.     If  the  time 
selected  by  the  one  in  Paris  is  9  P.M.,  at  what  time  by  the 
clocks  in  Rome  must  the  other  person  begin  reading  the  poem  ? 

A  system  of  standard  time  has  been  adopted  in  the  United  States  by 
which  the  difference  in  time  between  places  differs  by  whole  hours  or 
not  at  all.  The  meridians  60°,  75°,  90°,  105°  and  120°  west  from  Green- 
wich are  called  time  meridians.  Places  within  7£°  east  or  7£°  west  of  the 
meridian  of  75°  have  tlte  time  of  that  meridian,  which  is  called  Eastern 
Time.  The  time  within  7|°  either  side  of  the  meridian  of  90°  is  called 
Central  Time.  The  time  within  7|°  either  side  of  the  meridian  of  105° 
is  Mountain  Time.  The  time  within  7£°  either  side  of  the  meridian  of 
120°  is  Pacific  Time. 

428.  How  many  hours  by  standard  time  are  there  between 
places  which  have  Eastern  Time  and  those  which  have  Moun- 
tain Time  ?     Pacific  Time  ?     Central  Time  ? 

429.  If  school  begins  at  9  o'clock,  how  long  have  the  chil- 
dren in  Denver  been  in  school  when  the  morning  session  begins 
on  the  Pacific  coast  ?     How  long  have  the  children  in  Boston 
been  in  school  ?     The  children  in  Chicago  ? 

430.  If  the  afternoon  session  begins  at  1  P.M.  and  closes  at 
3  P.M.,  in  what  part  of  the  country  is  the  afternoon  session 
just   beginning  when  the  children   in  Washington,  D.C.,  are 
being  dismissed  ? 

431.  A  train  entered  a  city  at  10  P.M.  Central  Time.    After  a 
stop  of  10  min.  the  train  left  the  city  at  9  hr.  10  min.  Mountain 
Time.     In  what  direction  was  the  train  running  ?     What  was 
the  longitude  of  the  city  ? 

432.  Is  the  present  time  of  Boston  slower  or  faster  than  the 
old  local  time  ? 


412  MEASUREMENTS  AND  CONSTRUCTIONS 


MISCELLANEOUS   EXERCISES 

.  simplify  m    m    m    m    m. 

100  50 


2.  Write  a  complex  fraction  in  which  each  number  in  the 
numerator  is  prime  and  each  number  in  the  denominator  is 
composite.     Simplify  it. 

3.  Make  a  fraction  whose  numerator  is  the  only  prime 
between  90  and  100,  and  whose  denominator  is  the  product 
of  all  the  primes  between  80  and  90.     Express  that  fraction  as 
per  cent. 

Which  is  greater,  and  how  much  : 

4.  3V441  or  V3721  ?  5.      5V676  or  13V121? 

6.  What  number  squared  equals  77,284  ? 

7.  Express  2  mi.  20  rd.  3  yd.  as  yards.     As  rods.     As 
miles. 

8.  At  7^  a  foot,  what  is  the  cost  of  2£  mi.  of  telephone 
wire? 

9.  At  7  /  a  foot,  how  much  will  it  cost  to  fence  a  square  lot 
containing  1521  sq.  rd.  ? 

10.    If    a  =  4,    how   much   is    cc2?     V#?     5V#?     V9x? 
7V25z?     aV64?     V21  +  a?     VoT+45? 

11  .   If  x  =  9,  and  y  =  16,  how  much  is  V#  +  \A/  ? 

Solve. 

12.    8:ar  =  aj:2.  13.    9  :  a;  =  a?:  16. 

Write  mean  proportionals  between  the  following  numbers 

14.  2  and  32.  19.      2J  and  10. 

15.  2  and  18.  20.    12J  and  50. 

16.  2  and  50.  21.      8£  and  75. 

17.  8  and  4J.  22.      6J  and  100. 

18.  3  and  27.  23.        f  and  54. 


MISCELLANEOUS  EXERCISES  413 

24.  Name  all  the   demominations   in  the  table  of  metric 
linear  measure.     Of  metric  square  measure.     Of  metric  cubic 
measure.     Of  metric  measure  of  capacity.     Of  metric  weight. 
Give  the  meaning  of  the  prefixes  in  the  metric  tables. 

25.  What  is  the  weight  of  a  cubic  centimeter  of  water  ?     Of 
a  liter  of  water  ? 

26.  A  tank  3  meters  long  and  2  meters  wide  is  filled  with 
water  to  the  depth  of  1£  meters.     How   many  kiloliters  of 
water  are  in  it  ? 

27.  How  many  meters  are  there  in  the  perimeter  of  a  right 
triangle  whose  base  is  27  centimeters  and  altitude  120  centim- 
eters ? 

28.  There  are  2400  square  decimeters  in  the  surface  of  a 
cube.     How  many  cubic  centimeters  does  it  contain  ? 

29.  Find  the  cost  of  digging  a  cellar  2  dekameters  long,  1£ 
dekameters  wide,  and  6  meters  deep  at  10  ^  a  cubic  meter. 

30.  The  measure  of  a  meter  was  found  by  taking  as  nearly 
as  possible  1 0  0  0*0  0  0  0  of  the  distance  from  the  equator  to  the 
pole.     When  a  man  has  traveled  one  kilometer  north  from  the 
equator,  how  far  is  he  from  each  pole  ? 

31.  What  are  the  dimensions  of  a  cube  that  holds  a  milli- 
liter?     A  kiloliter  ? 

32.  What  is  the  unit  of  land  measure  in  the  metric  system  ? 
What  are  its  dimensions  ? 

33.  One  side  of  a  piece  of  land  in  the  form  of  a  right  tri- 
angle is  2.7  kilometers,  the  side  perpendicular  to  it  is  3.6  kilo- 
meters.    What  is  the  value  of  the  land  at  $  50  a  hectare  ? 
What  is  the  cost  of  fencing  the  land  at  20  $  a  meter  ? 

34.  Give  the  dimensions  of  the  unit  of  wood  measure  in  the 
metric  system. 

35.  At  $1.50  per  stere,  what  is  the  value  of  a  pile  of  wood 
8  m.  long,  4  m.  wide,  and  2£  m.  high  ? 

36.  A  circle  is  inscribed  in  a  square  whose  side  is  10£  in. 
Find  the  area  of  the  circle. 


414  MEASUREMENTS  AND   CONSTRUCTIONS 

37.  Write  two  fractions,  such  that  the  quotient  of  the  greater 
divided  by  the  less  is  f . 

38.  Add  3  to  each  term  of  a  proper  fraction  and  determine 
whether  the  resulting  fraction  is  greater  or  less  than  the  original 
fraction. 

39.  Add  3  to  each  term  of  an  improper  fraction  and  deter- 
mine whether  the  resulting  fraction  is  greater  or  less  than  the 
original  fraction. 

40.  Name  three  perfect  squares  whose  sum  is  14.     21.     26. 
29.     30.     35.     38.     42. 

41.  Name  two  perfect  cubes  whose  sum  is  9.     35.    65.    133. 

42.  The  Kohinoor  diamond  weighs  103  carats.     The  Star  of 
Brazil  weighs  125  carats.     The  smaller  equals  what  per  cent  of 
the  larger  ?     The  larger  equals  what  per  cent  of  the  smaller  ? 

43.  Virginia  tobacco  contains  1%  nicotine.     How  much  nic- 
otine in  a  ton  of  it  ? 

44.  Some  India  rubber  was  bought  for  12£^  a  pound  where 
it  was  grown,  and  was  sold  at  50^  a  pound  in  this  country. 
What  was  the  per  cent  of  increase  in  price? 

45.  How  many  degrees  in  an  arc  which  is  40%  of  a  circum- 
ference?    60%  of  a  circumference  ?     50%  of  a  quadrant? 

46.  If  an  arc  of  45°  is  17  in.  long,  how  long  is  the  whole 
circumference  ? 

47.  Find  the  number  of  which  40  is  f  %. 

48.  120  rd.   are  8£%  of  how  many  rods?     12£%?     6J%? 

49.  If  the  decoration  of  a  building  required  3000  cu.  yd.,  25 
cu.  ft  of  stone,  and  that  was  7%  of  the  whole  amount  used  in 
constructing  the  building,  how  much  was  used  ? 

50.  Draw  four  isosceles  right  triangles  whose  equal  sides 
are  each  6  in.     Arrange  them  so  as  to  form  a  square.     An 
oblong.     A  right  triangle.      A  rhomboid.      A  trapezoid.     An 
irregular  pentagon.     An   irregular   hexagon.      Find   the  per- 
imeter and  area  of  each  figure. 


MISCELLANEOUS   EXERCISES  415 

51.  If  the  rainfall  on  a  certain  day  were  -J  of  an  inch,  how 
many  gallons  of  water  would  fall  on  an  acre  of  land  ? 

52.  A  boat  can  sail  15  mi.  an  hour  down  the  river,  and  10  mi. 
an  hour  up  the  river.      If  it  sails  down  the  river  for  2  hr.  and 
then  returns,  how  long  a  time  will  elapse  between  its  departure 
and  its  arrival  ? 

53.  $  1500  equal  25%  more  than  A's  money,  and  25%  less 
than  B's.     How  much  more  has  B  than  A? 

54.  A  box  28  in.  by  18  in.  by  14  in.  is  filled  with  packages 
of  coffee,  each  7  in.  by  3J  in.  by  3  in.     Find  the  value  of  all 
at  10^  per  package. 

55.  If  5  boys  do  a  piece  of  work  in  8  hr.,  how  long  will  it 
take  a  man  who  works  twice  as  fast  as  a  boy  ? 

56.  A  dealer  bought  a  number  of  stoves  for  $  240,  paying 
the  same  amount  for  each.     If  he  had  bought  another  dozen  of 
stoves  at  the  same  price  the  cost  of  both  invoices  of  stoves 
would  have  been  $  360.     How  much  did  each  stove  cost  ? 

57.  Mr.  Dean  earns  $1.25  per  day  and  his  brother  $1.75 
per  day.    .How  many  days  more  are  required  for  Mr.  Dean  to 
earn  $  350  than  for  his  brother  to  earn  it  ? 

58.  A  man  bought  a  cow  for  $35,  a  horse  for  3J  times  as 
much  as  the  cow,  and  a  wagon  for  $  1  more  than  ^  of  the  cost 
of  the  cow  and  the  horse.     How  much  was  paid  for  both  horse 
and  wagon? 

59.  A  man  and  his  wife  received  $  270  each  year  from  money 
which  they  had  at  interest.     The  man  received  $  30  more  than 
3  times  as  much  as  his  wife.     How  much  did  each  receive  ? 

60.  The  sum  of  $24,000  was  divided  between  A,  B,  and  C 
so  that  A  received  f  as  much  as  B,  and  C  $  4000  less  than  A 
and  B  together.     How  much  did  each  receive  ? 

61.  A  man  has  two  fields,  containing  10  A.  and   12 £  A. 
respectively.    Find  the  length,  in  rods,  of  the  side  of  a  square 
field  equal  in  area  to  both  fields. 


416  MEASUREMENTS  AND   CONSTRUCTIONS 

62.  Construct,  if  possible,  triangles  whose  sides  have  the 
following  lengths.     Measure  their  angles  with  a  protractor, 
and  tell  whether  each  triangle  is  right,  acute  angled,  or  obtuse 
angled. 

(a)  2  in.,  3  in.,  4  in.     (6)  2  in.,  4  in.,  5  in.       (c)  3  in.,  4  in.,  5  in. 
(d)  3  in.,  4  in.,  8  in.     (e)  3  in.,  4  in.,  7  in.       (/)  3  in.,  4  in.,  6  in. 

63.  Three  lines  being  given,  when  is  it  impossible  to  con- 
struct a  triangle  with  them  ? 

64.  Can  a  pyramid  be  constructed  whose  base  is  8  in.  square 
and  whose  slant  height  is  3  in.  ?     Explain. 

65.  A  toy  table  10  in.  high  is  an  exact  model  of  a  study 
table  30  in.  high.     If  a  leg  of  the  large  table  is  191  in.  long, 
how  long  is  a  leg  of  the  small  table  ?     If  the  area  of  the  top 
of  the  small  table  is  120  sq.  in.,  what  is  the  area  of  the  top  of 
the  large  table  ?     If  a  drawer  of  the  small  table  contains  40 
cu.  in.,  how  many  cubic  inches  does  the  corresponding  drawer 
of  the  large  table  contain  ? 

66.  A  train  running  at  the  rate  of  40  mi.  an  hour  starts 
from  Newburg,  to  go  to  Ironton,  a  distance  of  245  mi.     At  the 
same  time  another  train  going  30  mi.  an  hour  starts  from  Iron- 
ton  to  go  to  Newburg.     In  how  many  hours  will  they  meet  ? 

67.  Atmospheric  pressure  is  computed  to  be  15  Ib.  to  the 
square  inch.     At  that  rate  how  many  pounds  of  pressure  are 
upon  the  top  of  a  round  table  15  in.  in  diameter  ? 

68.  Suppose  Fig.  38,  page  401,  to  represent  the  upper  sur- 
face of  a  cheese  16  in.  in  diameter  and  6  in.  high,  from  which 
a  part  has  been  cut  whose  upper  surface  is  represented  by  the 
sector  AOB.     How  many  cubic  inches  are  in  the  part  that  is 
left? 

69.  A  cylindrical  tank  If  ft.  in  diameter  and  6  ft.  high  is 
half  full  of  water.    Assuming  that  a  cubic  foot  of  water  weighs 
1000  oz.,  how  many  pounds  of  wajfrer  are  in  the  tank  ? 


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